{"id":4621,"date":"2026-02-08T21:51:10","date_gmt":"2026-02-08T21:51:10","guid":{"rendered":"https:\/\/mrenglishkj.com\/?p=4621"},"modified":"2026-03-19T16:21:36","modified_gmt":"2026-03-19T16:21:36","slug":"sat-test-4-math-module-1st-solutions","status":"publish","type":"post","link":"https:\/\/us.mrenglishkj.com\/sat\/sat-test-4-math-module-1st-solutions\/","title":{"rendered":"SAT Math Test 4 Module 1st with Easy Solution &amp; Hack"},"content":{"rendered":"\n<h2 class=\"wp-block-heading\">Prepare For The SATs: Math Test Module 1st with Simple Explanations, Tips and Tricks<\/h2>\n\n\n\n<p>After giving three tests in a row, I am sure you will apply that learning here. This is the 4th version for you to practice your SAT Math learning. Like the other exams, it has the same format and all the necessary features for you to become a master in math. You just take the SAT Test Module First to practice your skills. The best part is that you practice within the time limit, and there are explanations of the correct answers and tips and tricks to get a perfect score on the SAT. You will find Math easy after step-by-step explanations.<\/p>\n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">ABOUT THE SAT MODULES<\/h3>\n\n\n\n<p>The SAT is divided into four modules. There are two categories with each split into two modules. The first category is &#8220;Reading and Writing&#8221; with two modules. The second category is &#8220;Math&#8221; with two modules. The one, you will do below is SAT Math Practice Test Module 1st.<\/p>\n\n\n\n<p>The first module has questions ranging from easy to difficult, but the second module only contains difficult questions. If you want to take some other SATs, visit the links below.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/us.mrenglishkj.com\/sat\/category\/sat-english\/module-1st\/\" target=\"_blank\" rel=\"noopener\" title=\"\">1st Module of SAT Reading And Writing Practice Tests<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/us.mrenglishkj.com\/sat\/category\/sat-english\/module-2nd\/\" target=\"_blank\" rel=\"noopener\" title=\"\">2nd Module of SAT Reading And Writing Practice Tests<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/us.mrenglishkj.com\/sat\/category\/sat-math\/1st-module\/\" target=\"_blank\" rel=\"noopener\" title=\"\">1st Module of SAT Math Practice Tests<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/us.mrenglishkj.com\/sat\/category\/sat-math\/2nd-module\/\" target=\"_blank\" rel=\"noopener\" title=\"\">2nd Module of SAT Math Practice Tests<\/a><\/li>\n<\/ul>\n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">THE SAT MATH MODULE 1ST<\/h3>\n\n\n\n<p>The first module of Math in SAT contains four segments: Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry. The questions in Module 1st are from easy to difficult. In a real SAT exam, you must answer 22 questions within 35 minutes. We have provided you with the same in this Practice Test.<\/p>\n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h4 class=\"wp-block-heading\">Instructions for the SAT Real-Time Exam<\/h4>\n\n\n\n<ol class=\"wp-block-list\">\n<li><strong>Go Back-and-Forth:<\/strong> You will see an arrow on the right or left corner of the slide, click to move forward or backward.<\/li>\n\n\n\n<li><strong>Interaction:<\/strong> You will see a press button at the top right corner that tells you there are some interactive components in the slide. Click the press button to find out.<\/li>\n\n\n\n<li><strong>Timer: <\/strong>On the top of the slide, you will see the timer, we have divided the time based on the average of the module 1st. (The 35 minutes are equally divided into 22 questions&#8217; time.) It is best to note the time before and after finishing the practice test to measure, &#8220;Was it within 35 minutes or not?&#8221;<\/li>\n\n\n\n<li><strong>Mute:<\/strong> You can click on the speaker button to mute the audio.<\/li>\n\n\n\n<li><strong>Image:<\/strong> You can click on a graph, table, or other image to expand it and view it in full screen.<\/li>\n\n\n\n<li><strong>Mobile:<\/strong> You cannot take the real exam on mobile, but our practice exam you can give on mobile.<\/li>\n\n\n\n<li><strong>Calculator<\/strong>: Below the Test, you will see a Desmos calculator and graph for Math. The same, Desmos, will be used in real exams, so learn &#8220;How to use Desmos Calculator.&#8221;<\/li>\n\n\n\n<li><strong>Tips:<\/strong> This article will help you learn more about the SAT Exams. <a href=\"https:\/\/us.mrenglishkj.com\/sat\/everything-about-the-sat\/\" target=\"_blank\" rel=\"noopener\" title=\"SAT: EVERYTHING ABOUT THE SAT\">SAT: EVERYTHING ABOUT THE SAT<\/a><\/li>\n<\/ol>\n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<figure class=\"wp-block-embed alignfull is-type-wp-embed is-provider-genially wp-block-embed-genially\"><div class=\"wp-block-embed__wrapper\">\n<iframe class=\"wp-embedded-content\" sandbox=\"allow-scripts\" security=\"restricted\" title=\"SAT 4 Math Module 1st\" frameborder='0' width='1200' height='675' src='https:\/\/view.genially.com\/6748a5b86ec982400c2f9e7f#?secret=gsmb2iHH16' data-secret='gsmb2iHH16' scrolling='yes'><\/iframe>\n<\/div><figcaption class=\"wp-element-caption\">Wait here for the SAT Test to appear.<\/figcaption><\/figure>\n\n\n\n<div style=\"height:40px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<p>Our team has reviewed some of the best SAT learning materials for your convenience. These materials are best for your career growth.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Check Our Review Blog: <a href=\"https:\/\/review.mrenglishkj.com\/\" target=\"_blank\" rel=\"noopener\" title=\"\">review.mrenglishkj.com<\/a><\/li>\n<\/ul>\n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<!-- HTML for the Desmos Calculator Embed (Always Visible) -->\n<div id=\"desmos-container\">\n    <iframe loading=\"lazy\"\n        src=\"https:\/\/www.desmos.com\/calculator\/fxgemyy2gl\"\n        width=\"100%\"\n        height=\"500px\"\n        frameborder=\"0\"\n        allowfullscreen\n    ><\/iframe>\n<\/div>\n\n<!-- Button to Open Calculator in Slide-Out Panel -->\n<button id=\"desmos-toggle\" style=\"position: fixed; top: 20px; right: 20px; z-index: 1000;\">\n    Open Calculator\n<\/button>\n\n<!-- Slide-Out Desmos Calculator Panel (hidden initially) -->\n<div id=\"desmos-panel\">\n    <iframe loading=\"lazy\"\n        src=\"https:\/\/www.desmos.com\/calculator\/fxgemyy2gl\"\n        width=\"100%\"\n        height=\"95%\"\n        frameborder=\"0\"\n        allowfullscreen\n    ><\/iframe>\n<\/div>\n\n<!-- CSS Styling for the Slide-Out Panel -->\n<style>\n    \/* Main Container Styling *\/\n    #desmos-container {\n        max-width: 600px; \/* Adjust as needed *\/\n        margin: 20px auto;\n    }\n\n    \/* Slide-Out Panel Styling *\/\n    #desmos-panel {\n        position: fixed;\n        top: 0;\n        right: -400px; \/* Hidden by default *\/\n        width: 400px; \/* Adjust width as needed *\/\n        height: 100vh;\n        background-color: white;\n        border-left: 1px solid #ccc;\n        box-shadow: -2px 0 5px rgba(0, 0, 0, 0.2);\n        transition: right 0.3s ease;\n        z-index: 999; \/* Ensure it overlays content *\/\n    }\n\n    #desmos-panel.open {\n        right: 0;\n    }\n<\/style>\n\n<!-- JavaScript to Toggle the Slide-Out Panel -->\n<script>\n    document.getElementById(\"desmos-toggle\").onclick = function() {\n        var panel = document.getElementById(\"desmos-panel\");\n        if (panel.classList.contains(\"open\")) {\n            panel.classList.remove(\"open\");\n        } else {\n            panel.classList.add(\"open\");\n        }\n    };\n<\/script>\n\n\n\n<p class=\"has-text-align-center has-small-font-size\">Wait for the Desmos Calculator to appear.<\/p>\n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h3 class=\"wp-block-heading\">SAT MATH PROBLEM SOLUTIONS WITH STEP-BY-STEP EXPLANATION<\/h3>\n\n\n\n<p>Do not open the tabs before finishing the practice test above! For your convenience, we have compiled all the solutions and their explanations here. We will also give you some tips or advice to help you understand them better. You&#8217;ll see <strong>&#8216;why this answer is correct&#8217;<\/strong> and <strong>&#8216;why this is incorrect.&#8217;<\/strong><\/p>\n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<h4 class=\"wp-block-heading\">Math Solutions and Explanations:<\/h4>\n\n\n\n<p>The light red color shows the Question, green shows the Correct answer, red shows the Incorrect one, and blue shows Tips or Tricks with step-by-step explanations.<\/p>\n\n\n\n<div class=\"wp-block-coblocks-accordion alignfull\">\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>1st Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"233\" height=\"236\" src=\"https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/11\/image-93.png\" alt=\"\" class=\"wp-image-4633\"\/><\/figure>\n\n\n\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> What is the y-intercept of the graph shown?<br>A) (\u22128, 0)<br>B) (\u22126, 0)<br>C) (0, 6)<br>D) (0, 8)<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Choice D<\/strong> is correct. The y-intercept of a graph in the xy-plane is the point at which the graph crosses the y-axis. The graph shown crosses the y-axis at the point (0, 8). Therefore, the y-intercept of the graph shown is (0, 8).<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice A is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice B is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice C is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Step 1: Understanding the y-intercept<\/strong><br>The <strong>y-intercept<\/strong> is the point where the graph intersects the <strong>y-axis<\/strong>.<br>~ On the y-axis, <em>x<\/em> = 0, so the y-intercept is the value of <em>y<\/em> when <em>x<\/em> = 0.<br>~ This means we are looking for the coordinate of the form (0, y) on the graph.<br><br><strong>Step 2: Observing the graph at <em>x<\/em> = 0<\/strong><br>~ From the graph provided, when <em>x<\/em> = 0, the value of <em>y<\/em> is <strong>clearly at 8<\/strong>.<br>~ The corresponding point is (0, 8).<br><br><strong>Step 3: Choosing the correct option<\/strong><br>~ The y-intercept is (0, 8).<br>~ Among the given options:<br>~ ~ <strong>Option A<\/strong>: (\u22128, 0) = Incorrect; this is not on the y-axis.<br>~ ~ <strong>Option B<\/strong>: (\u22126, 0) = Incorrect; this is also not on the y-axis.<br>~ ~ <strong>Option C<\/strong>: (0, 6) = Incorrect; the graph does not pass through <em>y<\/em> = 6 when <em>x<\/em> = 0.<br>~ ~ <strong>Option D<\/strong>: (0, 8) = Correct; this matches the point where the graph intersects the y-axis.<br>Thus, <strong>Option D<\/strong> is the correct answer.<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>2nd Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> The total cost <em>f<\/em>(x), in dollars, to lease a car for 36 months from a particular car dealership is given by f(x) = 36x + 1,000, where <em>x<\/em> is the monthly payment, in dollars. What is the total cost to lease a car when the monthly payment is dollar sign $400?<br>A) $13,400<br>B) $13,000<br>C) $15,400<br>D) $37,400<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Choice C <\/strong>is correct. It\u2019s given that f(x) is the total cost, in dollars, to lease a car from this dealership with a monthly payment of x dollars. Therefore, the total cost, in dollars, to lease the car when the monthly payment is $400 is represented by the value of f(x) when x = 400. Substituting 400 for x in the equation f(x) = 36x + 1,000 yields f(400) = 36(400) + 1,000, or f(400) = 15,400. Thus, when the monthly payment is $400, the total cost to lease a car is $15,400.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice A is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice B is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice D is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Step 1: Understand the function and given values<\/strong><br>The function is provided as:<br><em>f<\/em>(x) = 36<em>x<\/em> + 1,000<br>~ <em>f<\/em>(x) represents the <strong>total cost<\/strong> of leasing the car for 36 months, in dollars.<br>~ <em>x<\/em> represents the <strong>monthly payment<\/strong>, in dollars.<br>~ We are asked to find <em>f<\/em>(x) when <em>x<\/em> = 400 (monthly payment is $400).<br><br><strong>Step 2: Substitute <em>x<\/em> = 400 into the function<\/strong><br>To calculate the total cost <em>f<\/em>(x) when <em>x<\/em> = 400, substitute 400 for <em>x<\/em> in the function:<br><em>f<\/em>(x) = 36<em>x<\/em> + 1,000<br><em>f<\/em>(400) = 36(400) +1,000<br><br><strong>Step 3: Perform the calculations<\/strong><br>~ Multiply 36 \u00d7 400:<br>36 \u00d7 400 = 14,400<br>~ Add 1,000 to the result:<br><em>f<\/em>(400) = 14,400 + 1,000 = 15,400<br><br><strong>Step 4: State the final answer<\/strong><br>The total cost to lease the car when the monthly payment is $400 is: <strong>15,400&nbsp;dollars.<\/strong>\u200b<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>3rd Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> Each side of a square has a length of 45. What is the perimeter of this square?<br>A) 45<br>B) 180<br>C) 90<br>D) 360<br>[Type-Based Answer: In the final exam, you will type the answer rather than choose from options.]<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Option B:<\/strong> The correct answer is 180. The perimeter of a polygon is equal to the sum of the lengths of the sides of the polygon. It\u2019s given that each side of the square has a length of 45. Since a square is a polygon with 4 sides, the perimeter of this square is 45 + 45 + 45 + 45, or 180.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice A is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice C is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice D is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Step 1: Understand the problem<\/strong><br>The question gives:<br>~ The <strong>length of each side<\/strong> of a square = 45.<br>~ We need to find the <strong>perimeter<\/strong> of the square.<br>The formula for the perimeter of a square is:<br>Perimeter= 4 \u00d7 side&nbsp;length<br><br><strong>Step 2: Substitute the side length<\/strong><br>Substitute 45 for the side length in the formula: Perimeter= 4 \u00d7 45<br><br><strong>Step 3: Perform the calculation<\/strong><br>4 \u00d7 45 = 180<br><br><strong>Step 4: State the final answer<\/strong><br>The perimeter of the square is: <strong>180&nbsp;units<\/strong>\u200b<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>4th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"305\" height=\"64\" src=\"https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/11\/image-94.png\" alt=\"SAT Math question and answer explanation and learning\" class=\"wp-image-4639\" srcset=\"https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/11\/image-94.png 305w, https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/11\/image-94-300x63.png 300w\" sizes=\"auto, (max-width: 305px) 100vw, 305px\" \/><\/figure>\n\n\n\n<p class=\"is-style-warning\" style=\"font-size:0.9em\">A) 11<br>B) -11<br>C) 55<br>D) 5<br>[Type-Based Answer: In the final exam, you will type the answer rather than choose from options.]<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Option D: <\/strong>The correct answer is 5. Multiplying both sides of the given equation by x + 6 results in 55 = x(x + 6). Applying the distributive property of multiplication to the right-hand side of this equation results in 55 = x<sup>2<\/sup> + 6x. Subtracting 55 from both sides of this equation results in 0 = x<sup>2<\/sup> + 6x &#8211; 55. The right-hand side of this equation can be rewritten by factoring. The two values that multiply to<br>-55 and add to 6 are 11 and -5. It follows that the equation 0 = x<sup>2<\/sup> + 6x &#8211; 55 can be rewritten as 0 = (x + 11)(x &#8211; 5). Setting each factor equal to 0 yields two equations: x + 11 = 0 and x &#8211; 5 = 0. Subtracting 11 from both sides of the equation x + 11 = 0 results in x = -11. Adding 5 to both sides of the equation x &#8211; 5 = 0 results in x = 5. Therefore, the positive solution to the given equation is 5.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice A is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice B is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice C is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\">We are tasked with finding the positive solution to this equation.<br><strong>Step 1: Eliminate the fraction<\/strong><br>Multiply both sides of the equation by <em>x<\/em> + 6 (the denominator) to eliminate the fraction. This step ensures that we don&#8217;t divide by zero later.<br>55 = <em>x<\/em>(<em>x<\/em> + 6)<br><br><strong>Step 2: Expand the equation<\/strong><br>Expand the right-hand side: 55 = x<sup>2<\/sup> + 6x<br><br><strong>Step 3: Rearrange into standard quadratic form<\/strong><br>Move all terms to one side of the equation to set it equal to 0:<br>x<sup>2<\/sup> + 6x \u2212 55 = 0<br><br><strong>Step 4: Solve the quadratic equatio<\/strong>n<br>The equation x<sup>2<\/sup> + 6x \u2212 55 = 0 is a standard quadratic equation. We will solve it using the quadratic formula:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"563\" height=\"612\" src=\"https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/11\/image-96.png\" alt=\"SAT Math advanced math step-by-step solutions and practice\" class=\"wp-image-4642\" srcset=\"https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/11\/image-96.png 563w, https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/11\/image-96-276x300.png 276w\" sizes=\"auto, (max-width: 563px) 100vw, 563px\" \/><\/figure>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Step 5: Find the two solutions<\/strong><br>Now, calculate the two possible values of <em>x<\/em>:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"559\" height=\"587\" src=\"https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/11\/image-95.png\" alt=\"SAT step-by-step practice and learning for free\" class=\"wp-image-4641\" srcset=\"https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/11\/image-95.png 559w, https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/11\/image-95-286x300.png 286w\" sizes=\"auto, (max-width: 559px) 100vw, 559px\" \/><\/figure>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\">The solution is verified.<br><br><strong>Final Answer<\/strong><br>The positive solution is: 5\u200b<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>5th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> An object travels at a constant speed of 12 centimeters per second. At this speed, what is the time, in seconds, that it would take for the object to travel 108 centimeters?<br>A) 9<br>B) 96<br>C) 120<br>D) 972<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Choice A<\/strong> is correct. If the object travels 108 centimeters at a speed of 12 centimeters per second, the time of travel can be determined by dividing the total distance by the speed. This results in 108 centimeters divided by 12 centimeters\/second, which is 9 seconds.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice B is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice C is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice D is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Step-by-Step Solution<\/strong><br><strong>Step 1: Understand the relationship between speed, distance, and time<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"621\" height=\"591\" src=\"https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/11\/image-97.png\" alt=\"\" class=\"wp-image-4648\" srcset=\"https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/11\/image-97.png 621w, https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/11\/image-97-300x286.png 300w\" sizes=\"auto, (max-width: 621px) 100vw, 621px\" \/><\/figure>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Step 3: Perform the calculation<\/strong><br>Divide 108 by 12:<br>Time = 9\u2009seconds<br><br><strong>Step 4: Verify<\/strong><br>At a speed of 12 cm\/s, traveling for 9 seconds would result in:<br>Distance = Speed \u00d7 Time<br>Distance = 12 \u00d7 9<br>Distance = 108\u2009cm<br>The calculation checks out.<br><br><strong>Final Answer<\/strong>: <strong>9 seconds.<\/strong><\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>6th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> Right triangles \u25b3ABC and \u25b3DEF are similar, where <em>A<\/em> and <em>B<\/em> correspond to <em>D<\/em> and <em>E<\/em>, respectively. If \u2220B measures 45<sup>\u2218<\/sup>, what is the measure of \u2220E?<br>A) 35<sup>\u2218<\/sup><br>B) 45<sup>\u2218<\/sup><br>C) 55<sup>\u2218<\/sup><br>D) 90<sup>\u2218<\/sup><\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Choice B<\/strong> is correct. Since we know both \u25b3ABC and \u25b3DEF are similar and if \u2220B measures 45<sup>\u2218<\/sup> then you should focus on A = D, B = E, and C = F. Hence \u2220E is 45<sup>\u2218<\/sup>.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice A is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice C is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice D is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Step 1: Understand triangle similarity properties<\/strong><br>Since the two triangles, \u25b3ABC and \u25b3DEF are similar, their corresponding angles are congruent. This means:<br>\u2220B = \u2220E<br>Thus, if \u2220B measures 45<sup>\u2218<\/sup>, then \u2220E must also measure 45<sup>\u2218<\/sup>.<br><br><strong>Step 2: Verify other triangle angle properties<\/strong><br>In a triangle, the sum of all interior angles is always 180<sup>\u2218<\/sup>. For both triangles \u25b3ABC and \u25b3DEF, this holds true. Therefore:<br>~ In \u25b3ABC, the angles are \u2220A + \u2220B + \u2220C = 180<sup>\u2218<\/sup>.<br>~ Similarly, in \u25b3DEF, the angles are \u2220D + \u2220E + \u2220F = 180<sup>\u2218<\/sup>.<br>Since \u2220B = \u2220E, their measures are identical, confirming that \u2220E = 45<sup>\u2218<\/sup>.<br><br><strong>Final Answer:<\/strong> <strong>B)\u200945<sup>\u2218<\/sup><\/strong>\u200b<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>7th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> The function \ud835\udc54(\ud835\udc65) = 20 + 5\ud835\udc65 represents the total cost, in dollars, of attending a carnival when \ud835\udc65 rides are taken. How many rides can be taken for a total cost of $70?<br>A) 8 rides<br>B) 9 rides<br>C) 10 rides<br>D) 11 rides<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Choice C<\/strong> is correct.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice A is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice B is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice D is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Step 1: Understand the given function<\/strong><br>The function <em>g<\/em>(<em>x<\/em>) = 20 + 5<em>x<\/em> gives the total cost, where:<br>~ 20 is the fixed entry cost to attend the carnival.<br>~ 5<em>x<\/em> represents the cost of <em>x<\/em> rides, with each ride costing $5.<br>We are tasked with finding the number of rides (<em>x<\/em>) that can be taken for a total cost of $70.<br><br><strong>Step 2: Set up the equation<\/strong><br>Substitute the total cost into the equation <em>g<\/em>(<em>x<\/em>) = 70:<br>70 = 20 + 5<em>x<\/em><br><br><strong>Step 3: Solve for <em>x<\/em><\/strong><br>1. Subtract 20 from both sides to isolate the 5<em>x<\/em>-term:<br>70 \u2212 20 = 5<em>x<\/em><br>50 = 5<em>x<\/em><br>2. Divide both sides by 5 to solve for <em>x<\/em>:<br><em>x<\/em> = 50\/5<br><em>x<\/em> = 10<br><br><strong>Step 4: Verify the solution<\/strong><br>If <em>x<\/em> = 10, the total cost becomes:<br><em>g<\/em>(<em>x<\/em>) = 20 + 5(10)<br><em>g<\/em>(<em>x<\/em>) =20 + 50<br><em>g<\/em>(<em>x<\/em>) = 70<br>This matches the total cost given in the problem.<br><br><strong>Final Answer:<\/strong> <strong>C)\u200910&nbsp;rides<\/strong>\u200b<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>8th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> A bookstore has an initial inventory of 3,200 books. Each day, 50 books are sold. Based on this rate, in how many days will the inventory be reduced to 1,900 books?<br>A) 24 days<br>B) 26 days<br>C) 28 days<br>D) 32 days<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Choice B<\/strong> is correct.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice A is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice C is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice D is incorrect. This is the absolute value of y, not the value of <em>x<\/em>.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Step 1: Understand the given information<\/strong><br>The bookstore starts with an inventory of 3,200 books, and 50 books are sold each day. The task is to find how many days it will take for the inventory to reach 1,900 books.<br><br><strong>Step 2: Set up the equation<\/strong><br>The number of books left after <em>x<\/em> days can be expressed as:<br>Remaining&nbsp;books = Initial&nbsp;inventory \u2212 (Books&nbsp;sold&nbsp;per&nbsp;day \u00d7 Number&nbsp;of&nbsp;days)<br>Substituting the given information:<br>1,900 = 3,200 \u2212 50<em>x<\/em><br><br><strong>Step 3: Solve for <em>x<\/em><\/strong><br>1. Subtract 1,900 from both sides to isolate the 3,200 \u2212 50<em>x<\/em>\u2212term:<br>1,900 \u2212 3,200 = \u221250<em>x<\/em><br>-1,300 = -50<em>x<\/em><br>2. Divide both sides by \u221250 to solve for <em>x<\/em>:<br><em>x<\/em> = \u22121,300\/\u221250<br><em>x<\/em> = 26<br><br><strong>Step 4: Verify the solution<\/strong><br>If <em>x<\/em> = 26, the remaining books are:<br>Remaining&nbsp;books = 3,200 \u2212 (50 \u00d7 26)<br>Remaining books =3,200 \u2212 1,300<br>Remaining books = 1,900<br>This matches the target inventory.<br><br><strong>Final Answer:<\/strong> <strong>B)\u200926\u2009days<\/strong><\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>9th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong><em>y<\/em> &gt; 4<em>x<\/em> + 8<br>Question:<\/strong> For which of the following tables are all the values of x and their corresponding values of y solutions to the given inequality?<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"126\" height=\"347\" src=\"https:\/\/us.mrenglishkj.com\/sat\/sat\/wp-content\/uploads\/2024\/12\/image.png\" alt=\"Practice of the SAT test and learn SAT Math\" class=\"wp-image-4674\" srcset=\"https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/12\/image.png 126w, https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/12\/image-109x300.png 109w\" sizes=\"auto, (max-width: 126px) 100vw, 126px\" \/><\/figure>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Choice A<\/strong> is correct. In each choice, the values of <em>x<\/em> are 2, 4, and 6. Substituting the first value of <em>x<\/em>, 2, for <em>x<\/em> in the given inequality yields <em>y<\/em> &gt; 4(2) + 8, or <em>y<\/em> &gt; 16. Therefore, when <em>x<\/em> = 2, the corresponding value of <em>y<\/em> must be greater than 16. Of the given choices, only choice A is a table where the value of <em>y<\/em> corresponding to <em>x<\/em> = 2 is greater than 16. To confirm that the other values of <em>x<\/em> in this table and<br>their corresponding values of <em>y<\/em> are also solutions to the given inequality, the values of <em>x<\/em> and <em>y<\/em> in the table can be substituted for <em>x<\/em> and <em>y<\/em> in the given inequality. Substituting 4 for <em>x<\/em> and 30 for <em>y<\/em> in the given inequality yields 30 &gt; 4(4) + 8, or 30 &gt; 24, which is true. Substituting 6 for <em>x<\/em> and 41 for <em>y<\/em> in the<br>given inequality yields 41 &gt; 4(6) + 8, or 41 &gt; 32, which is true. It follows that for choice A, all the values of <em>x<\/em> and their corresponding values of <em>y<\/em> are solutions to the given inequality.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice B is incorrect. Substituting 2 for <em>x<\/em> and 8 for <em>y<\/em> in the given inequality yields 8 &gt; 4(2) + 8, or 8 &gt; 16, which is false.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice C is incorrect. Substituting 2 for <em>x<\/em> and 13 for <em>y<\/em> in the given inequality yields 13 &gt; 4(2) + 8, or 13 &gt; 16, which is false.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice D is incorrect. Substituting 2 for <em>x<\/em> and 13 for <em>y<\/em> in the given inequality yields 13 &gt; 4(2) + 8, or 13 &gt; 16, which is false.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Given Problem:<\/strong><br>We need to verify for which of the given tables (A, B, C, or D) all the <em>x<\/em> and corresponding <em>y<\/em> values satisfy the inequality:<br>y &gt; 4<em>x<\/em> + 8<br><br><strong>Step-by-Step Solution:<\/strong><br><strong>Step 1: Understand the inequality<\/strong><br>The inequality <em>y<\/em> &gt; 4<em>x<\/em> + 8 requires that, for each pair of <em>x<\/em> and <em>y<\/em>, the value of <em>y<\/em> must be <strong>strictly greater<\/strong> than the calculated value of 4<em>x<\/em> + 8.<br><br><strong>Step 2: Compute 4<em>x<\/em> + 8 for each <em>x<\/em> value in the tables<\/strong><br>We will substitute the <em>x<\/em> values from each table into 4<em>x<\/em> + 8, then compare the resulting values with the corresponding <em>y<\/em> values to check whether the inequality is satisfied.<br><br><strong>Option A:<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table is-style-regular\" style=\"font-size:0.8em\"><table class=\"has-fixed-layout\"><thead><tr><th class=\"has-text-align-center\" data-align=\"center\"><em>x<\/em><\/th><th class=\"has-text-align-center\" data-align=\"center\"><em>y<\/em><\/th><th class=\"has-text-align-center\" data-align=\"center\">4<em>x<\/em> + 8<\/th><th class=\"has-text-align-center\" data-align=\"center\">Check if<br><em>y<\/em> &gt; 4<em>x<\/em> + 8<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">2<\/td><td class=\"has-text-align-center\" data-align=\"center\">19<\/td><td class=\"has-text-align-center\" data-align=\"center\">4(2) + 8 = 16<\/td><td class=\"has-text-align-center\" data-align=\"center\">19 &gt; 16: True<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">4<\/td><td class=\"has-text-align-center\" data-align=\"center\">30<\/td><td class=\"has-text-align-center\" data-align=\"center\">4(4) + 8 = 24<\/td><td class=\"has-text-align-center\" data-align=\"center\">30 &gt; 24: True<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">6<\/td><td class=\"has-text-align-center\" data-align=\"center\">41<\/td><td class=\"has-text-align-center\" data-align=\"center\">4(6) + 8 = 32<\/td><td class=\"has-text-align-center\" data-align=\"center\">41 &gt; 32: True<\/td><\/tr><\/tbody><\/table><figcaption class=\"wp-element-caption\"><strong>Conclusion:<\/strong> All rows in <strong>Option A<\/strong> satisfy y &gt; 4<em>x<\/em> + 8.<\/figcaption><\/figure>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\">We have already got our correct option but still, let&#8217;s try solving other options for better clearance.<br><br><strong>Option B:<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table is-style-regular\" style=\"font-size:0.8em\"><table class=\"has-fixed-layout\"><thead><tr><th class=\"has-text-align-center\" data-align=\"center\"><em>x<\/em><\/th><th class=\"has-text-align-center\" data-align=\"center\"><em>y<\/em><\/th><th class=\"has-text-align-center\" data-align=\"center\">4<em>x<\/em> + 8<\/th><th class=\"has-text-align-center\" data-align=\"center\">Check if<br><em>y<\/em> &gt; 4<em>x<\/em> + 8<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">2<\/td><td class=\"has-text-align-center\" data-align=\"center\">8<\/td><td class=\"has-text-align-center\" data-align=\"center\">4(2) + 8 = 16<\/td><td class=\"has-text-align-center\" data-align=\"center\">8 &gt; 16: False<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">4<\/td><td class=\"has-text-align-center\" data-align=\"center\">16<\/td><td class=\"has-text-align-center\" data-align=\"center\">4(4) + 8 = 24<\/td><td class=\"has-text-align-center\" data-align=\"center\">16 &gt; 24: False<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">6<\/td><td class=\"has-text-align-center\" data-align=\"center\">24<\/td><td class=\"has-text-align-center\" data-align=\"center\">4(6) + 8 = 32<\/td><td class=\"has-text-align-center\" data-align=\"center\">24 &gt; 32: False<\/td><\/tr><\/tbody><\/table><figcaption class=\"wp-element-caption\"><strong>Conclusion:<\/strong> None of the rows in <strong>Option B<\/strong> satisfy y &gt; 4<em>x<\/em> + 8.<\/figcaption><\/figure>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Option C:<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table is-style-regular\" style=\"font-size:0.8em\"><table class=\"has-fixed-layout\"><thead><tr><th class=\"has-text-align-center\" data-align=\"center\"><em>x<\/em><\/th><th class=\"has-text-align-center\" data-align=\"center\"><em>y<\/em><\/th><th class=\"has-text-align-center\" data-align=\"center\">4<em>x<\/em> + 8<\/th><th class=\"has-text-align-center\" data-align=\"center\">Check if<br><em>y<\/em> &gt; 4<em>x<\/em> + 8<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">2<\/td><td class=\"has-text-align-center\" data-align=\"center\">13<\/td><td class=\"has-text-align-center\" data-align=\"center\">4(2) + 8 = 16<\/td><td class=\"has-text-align-center\" data-align=\"center\">13 &gt; 16: False<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">4<\/td><td class=\"has-text-align-center\" data-align=\"center\">18<\/td><td class=\"has-text-align-center\" data-align=\"center\">4(4) + 8 = 24<\/td><td class=\"has-text-align-center\" data-align=\"center\">18 &gt; 24: False<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">6<\/td><td class=\"has-text-align-center\" data-align=\"center\">23<\/td><td class=\"has-text-align-center\" data-align=\"center\">4(6) + 8 = 32<\/td><td class=\"has-text-align-center\" data-align=\"center\">23 &gt; 32: False<\/td><\/tr><\/tbody><\/table><figcaption class=\"wp-element-caption\"><strong>Conclusion:<\/strong> None of the rows in <strong>Option C<\/strong> satisfy y &gt; 4<em>x<\/em> + 8.<\/figcaption><\/figure>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Option D:<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-table is-style-regular\" style=\"font-size:0.8em\"><table class=\"has-fixed-layout\"><thead><tr><th class=\"has-text-align-center\" data-align=\"center\"><em>x<\/em><\/th><th class=\"has-text-align-center\" data-align=\"center\"><em>y<\/em><\/th><th class=\"has-text-align-center\" data-align=\"center\">4<em>x<\/em> + 8<\/th><th class=\"has-text-align-center\" data-align=\"center\">Check if<br><em>y<\/em> &gt; 4<em>x<\/em> + 8<\/th><\/tr><\/thead><tbody><tr><td class=\"has-text-align-center\" data-align=\"center\">2<\/td><td class=\"has-text-align-center\" data-align=\"center\">13<\/td><td class=\"has-text-align-center\" data-align=\"center\">4(2) + 8 = 16<\/td><td class=\"has-text-align-center\" data-align=\"center\">13 &gt; 16: False<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">4<\/td><td class=\"has-text-align-center\" data-align=\"center\">21<\/td><td class=\"has-text-align-center\" data-align=\"center\">4(4) + 8 = 24<\/td><td class=\"has-text-align-center\" data-align=\"center\">21 &gt; 24: False<\/td><\/tr><tr><td class=\"has-text-align-center\" data-align=\"center\">6<\/td><td class=\"has-text-align-center\" data-align=\"center\">29<\/td><td class=\"has-text-align-center\" data-align=\"center\">4(6) + 8 = 32<\/td><td class=\"has-text-align-center\" data-align=\"center\">29 &gt; 32: False<\/td><\/tr><\/tbody><\/table><figcaption class=\"wp-element-caption\"><strong>Conclusion:<\/strong> None of the rows in <strong>Option D<\/strong> satisfy y &gt; 4<em>x<\/em> + 8.<\/figcaption><\/figure>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Final Answer:<\/strong><br>The correct option is <strong>A<\/strong>, where all the <em>x<\/em> and <em>y<\/em> values satisfy the inequality <em>y<\/em> &gt; 4<em>x<\/em> + 8.<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>10th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> Which expression is equivalent to (<em>x<\/em><sup>2<\/sup> + 11)<sup>2<\/sup> + (<em>x<\/em> \u2212 5)(<em>x<\/em> + 5)?<br>A) <em>x<\/em><sup>4<\/sup> + 23<em>x<\/em><sup>2<\/sup> \u2212 14<br>B) <em>x<\/em><sup>4<\/sup> + 23<em>x<\/em><sup>2<\/sup> + 96<br>C) <em>x<\/em><sup>4<\/sup> + 12<em>x<\/em><sup>2<\/sup> + 121<br>D) x<sup>4<\/sup> + x<sup>2<\/sup> + 146<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Choice B<\/strong> is correct. The expression (x<sup>2<\/sup> + 11)<sup>2<\/sup> can be written as (<em>x<\/em><sup>2<\/sup> + 11)(<em>x<\/em><sup>2<\/sup> + 11), which is equivalent to <em>x<\/em><sup>2<\/sup>(<em>x<\/em><sup>2<\/sup> + 11) + 11(<em>x<\/em><sup>2<\/sup>  + 11). Distributing x<sup>2<\/sup> and 11 to (x<sup>2<\/sup> + 11) yields x<sup>4<\/sup> + 11x<sup>2<\/sup> + 11x<sup>2<\/sup> + 121. The<br>expression (<em>x<\/em> &#8211; 5)(<em>x<\/em> + 5) is equivalent to (<em>x<\/em> &#8211; 5)<em>x<\/em> + (<em>x<\/em> &#8211; 5)5. Distributing <em>x<\/em> and 5 to (<em>x<\/em> &#8211; 5) yields <em>x<\/em><sup>2<\/sup> &#8211; 5<em>x<\/em> + 5<em>x<\/em> &#8211; 25, or <em>x<\/em><sup>2<\/sup> &#8211; 25. Therefore, the expression (<em>x<\/em><sup>2<\/sup> + 11)<sup>2<\/sup> + (<em>x<\/em> &#8211; 5)(<em>x<\/em> + 5) is equivalent to (<em>x<\/em><sup>4<\/sup> + 22<em>x<\/em><sup>2<\/sup> + 121) + (<em>x<\/em><sup>2<\/sup> &#8211; 25), or x<sup>4<\/sup> + 22x<sup>2<\/sup> + 121 + x<sup>2<\/sup>  &#8211; 25. Combining like terms in this expression yields <em>x<\/em><sup>4<\/sup> + 23<em>x<\/em><sup>2<\/sup> + 96.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice A is incorrect. Equivalent expressions must be equivalent for any value of <em>x<\/em>. Substituting 0 for <em>x<\/em> in this expression yields -14, whereas substituting 0 for <em>x<\/em> in the given expression yields 96.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice C is incorrect. Equivalent expressions must be equivalent for any value of <em>x<\/em>. Substituting 0 for <em>x<\/em> in this expression yields 121, whereas substituting 0 for <em>x<\/em> in the given expression yields 96.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice D is incorrect. Equivalent expressions must be equivalent for any value of <em>x<\/em>. Substituting 0 for <em>x<\/em> in this expression yields 146, whereas substituting 0 for <em>x<\/em> in the given expression yields 96.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Step-by-Step Solution<\/strong><br><strong>Step 1: Expand (x<sup>2<\/sup> + 11)<sup>2<\/sup><\/strong><br>Using the formula for the square of a binomial, (<em>a<\/em> + <em>b<\/em>)<sup>2<\/sup> = <em>a<\/em><sup>2<\/sup> + 2<em>ab<\/em> + <em>b<\/em><sup>2<\/sup>, expand (x<sup>2<\/sup> + 11)<sup>2<\/sup>:<br>(x<sup>2<\/sup> + 11)<sup>2<\/sup> = (<em>x<\/em><sup>2<\/sup>)<sup>2<\/sup> + 2(<em>x<\/em><sup>2<\/sup>)(11) + 11<sup>2<\/sup>.<br>Simplify each term:<br>~ (<em>x<\/em><sup>2<\/sup>)<sup>2<\/sup> = <em>x<\/em><sup>4<\/sup>,<br>~ 2(<em>x<\/em><sup>2<\/sup>)(11) = 22<em>x<\/em><sup>2<\/sup><br>~ 11<sup>2<\/sup> = 121<br>Thus: (x<sup>2<\/sup> + 11)<sup>2<\/sup> = x<sup>4<\/sup> + 22<em>x<\/em><sup>2<\/sup> + 121.<br><br><strong>Step 2: Expand (x \u2212 5)(x + 5)<\/strong><br>Using the difference of squares formula, (<em>a<\/em> \u2212 <em>b<\/em>)(<em>a<\/em> + <em>b<\/em>) = <em>a<\/em><sup>2<\/sup> \u2212 <em>b<\/em><sup>2<\/sup>, expand (<em>x<\/em> \u2212 5)(<em>x<\/em> + 5):<br>(<em>x<\/em> \u2212 5)(<em>x<\/em> + 5) = <em>x<\/em><sup>2<\/sup> \u2212 5<sup>2<\/sup>.<br>Simplify: <em>x<\/em><sup>2<\/sup> \u2212 5<sup>2<\/sup> = <em>x<\/em><sup>2<\/sup> \u2212 25.<br><br><strong>Step 3: Add the expanded terms<\/strong><br>Now add the results from Steps 1 and 2:<br>(<em>x<\/em><sup>2<\/sup> + 11)<sup>2<\/sup> + (<em>x<\/em> \u2212 5)(<em>x<\/em> + 5)<br>(<em>x<\/em><sup>4<\/sup> + 22<em>x<\/em><sup>2<\/sup> + 121) + (<em>x<\/em><sup>2<\/sup> \u2212 25).<br>x<sup>4<\/sup> + 22x<sup>2<\/sup> + 121 + x<sup>2<\/sup> \u2212 25<br>x<sup>4<\/sup> + 22x<sup>2<\/sup> + x<sup>2<\/sup> + 121 \u2212 25<br>x<sup>4<\/sup> + 23x<sup>2<\/sup> + 96<br>Group-like terms:<br>~ <em>x<\/em><sup>4<\/sup>,<br>~ 22<em>x<\/em><sup>2<\/sup> + <em>x<\/em><sup>2<\/sup> = 23<em>x<\/em><sup>2<\/sup>,<br>~ 121 \u2212 25 = 96 (Positive 96 because 121 is a bigger value).<br>Thus: (<em>x<\/em><sup>2<\/sup> + 11)<sup>2<\/sup> + (<em>x<\/em> \u2212 5)(<em>x<\/em> + 5) = <em>x<\/em><sup>4<\/sup> + 23<em>x<\/em><sup>2<\/sup> + 96.<br><br><strong>Final Answer:<\/strong><br>The equivalent expression is: B) <em>x<\/em><sup>4<\/sup> + 23<em>x<\/em><sup>2<\/sup> + 96.<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>11th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> What is the equation of the line that passes through the point (0, 5) and is parallel to the graph of<br><em>y<\/em> = 7<em>x<\/em> + 4 in the <em>xy<\/em>-plane?<br>A) <em>y<\/em> = 5<em>x<\/em><br>B) <em>y<\/em> = 7<em>x<\/em> + 5<br>C) <em>y<\/em> = 7<em>x<\/em><br>D) <em>y<\/em> = 5<em>x<\/em> + 7<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Choice B<\/strong> is correct. The equation of a line in the <em>xy<\/em>-plane can be written in slope-intercept form <em>y = mx + b<\/em>, where <em>m<\/em> is the slope of the line and (0, <em>b<\/em>) is its <em>y<\/em>-intercept. It\u2019s given that the line passes through the point (0, 5). Therefore, <em>b<\/em> = 5. It\u2019s also given that the line is parallel to the graph of <em>y<\/em> = 7<em>x<\/em> + 4, which means the line has the same slope as the graph of <em>y<\/em> = 7<em>x<\/em> + 4. The slope of the graph of <em>y<\/em> = 7<em>x<\/em> + 4 is 7. Therefore, <em>m<\/em> = 7. Substituting 7 for <em>m<\/em> and 5 for <em>b<\/em> in the equation <em>y = mx + b<\/em> yields <em>y<\/em> = 7<em>x<\/em> + 5.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice A is incorrect. The graph of this equation passes through the point (0, 0), not (0, 5), and has a slope of 5, not 7.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice C is incorrect. The graph of this equation passes through the point (0, 0), not (0, 5).<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice D is incorrect. The graph of this equation passes through the point (0, 7), not (0, 5), and has a slope of 5, not 7.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\">We are tasked with finding the equation of a line that passes through the point (0, 5) and is parallel to the graph of <em>y<\/em> = 7<em>x<\/em> + 4.<br><br><strong>Step-by-Step Solution<\/strong><br><strong>Step 1: Understand the slope of parallel lines<\/strong><br>The slope of a line parallel to another line must be the same as the slope of the original line. The given line <em>y<\/em> = 7<em>x<\/em> + 4 is in slope-intercept form (<em>y<\/em> = <em>mx<\/em> + <em>b<\/em>), where <em>m<\/em> represents the slope.<br>From <em>y<\/em> = 7<em>x<\/em> + 4, the slope <em>m<\/em> = 7. Therefore, the equation of the new line will also have a slope of <em>m<\/em> = 7.<br><br><strong>Step 2: Write the general form of the new line<\/strong><br>The equation of a line with slope <em>m<\/em> = 7 can be written in slope-intercept form:<br><em>y<\/em> = 7<em>x<\/em> + b,<br>where <em>b<\/em> is the <em>y<\/em>-intercept.<br><br><strong>Step 3: Use the given point to find <em>b<\/em><\/strong><br>We know the line passes through the point (0, 5). Substituting <em>x<\/em> = 0 and <em>y<\/em> = 5 into the equation <em>y<\/em> = 7<em>x<\/em> + <em>b<\/em>, we solve for <em>b<\/em>:<br>5 = 7(0) + <em>b<\/em>.<br>Simplify: <em>b<\/em> = 5.<br>Thus, the equation of the line is:<br><em>y<\/em> = 7x + 5.<br><br><strong>Step 4: Verify parallelism<\/strong><br>To confirm, the slope of the new line is <em>m<\/em> = 7, which is the same as the slope of the given line. Therefore, the two lines are indeed parallel.<br><br><strong>Final Answer:<\/strong> B)&nbsp;<em>y<\/em> = 7<em>x<\/em> + 5.\u200b<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>12th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> In the linear function <em>h<\/em>, <em>h<\/em>(0) = 41 and <em>h<\/em>(1) = 40. Which equation defines <em>h<\/em>?<br>A) <em>h<\/em>(<em>x<\/em>) = \u2212<em>x<\/em> + 41<br>B) <em>h<\/em>(<em>x<\/em>) = \u2212<em>x<\/em><br>C) <em>h<\/em>(<em>x<\/em>) = \u221241<em>x<\/em><br>D) <em>h<\/em>(<em>x<\/em>) = \u221241<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Choice A<\/strong> is correct. An equation defining a linear function can be written in the form <em>h<\/em>(x) = <em>ax<\/em> + <em>b<\/em>, where <em>a<\/em> and <em>b<\/em> are constants. It\u2019s given that <em>h<\/em>(0) = 41. Substituting 0 for <em>x<\/em> and 41 for <em>h<\/em>(x) in the equation <em>h<\/em>(x) = <em>ax<\/em> + <em>b<\/em> yields 41 = <em>a<\/em>(0) + <em>b<\/em>, or <em>b<\/em> = 41. Substituting 41 for <em>b<\/em> in the equation <em>h<\/em>(x) = <em>ax<\/em> + <em>b<\/em> yields <em>h<\/em>(<em>x<\/em>) = <em>ax<\/em> + 41. It\u2019s also given that <em>h<\/em>(1) = 40. Substituting 1 for <em>x<\/em> and 40 for <em>h<\/em>(<em>x<\/em>) in the equation <em>h<\/em>(x) = <em>ax<\/em> + 41 yields 40 = <em>a<\/em>(1) + 40, or 40 = <em>a<\/em> + 41. Subtracting 41 from the left and right-hand sides of this equation yields -1 = <em>a<\/em>. Substituting -1 for <em>a<\/em> in the equation <em>h<\/em>(x) = <em>ax<\/em> + 41 yields <em>h<\/em>(x) = -1<em>x<\/em> + 41, or <em>h<\/em>(x) = <em>-x<\/em> + 41.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice B is incorrect. Substituting 0 for <em>x<\/em> and 41 for <em>h<\/em>(<em>x<\/em>) in this equation yields 41 = -0, which isn\u2019t a true statement.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice C is incorrect. Substituting 0 for <em>x<\/em> and 41 for <em>h<\/em>(<em>x<\/em>) in this equation yields 41 = -41(0), or 41 = 0, which isn\u2019t a true statement.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice D is incorrect. Substituting 41 for <em>h<\/em>(<em>x<\/em>) in this equation yields 41 = -41, which isn\u2019t a true statement.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Step 1: Understand the structure of a linear equation<\/strong><br>A linear function is typically written in the form:<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"506\" height=\"671\" src=\"https:\/\/us.mrenglishkj.com\/sat\/sat\/wp-content\/uploads\/2024\/12\/image-2.png\" alt=\"Linear equation of the SAT math  solutions and explanations and learning\" class=\"wp-image-4703\" srcset=\"https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/12\/image-2.png 506w, https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/12\/image-2-226x300.png 226w\" sizes=\"auto, (max-width: 506px) 100vw, 506px\" \/><\/figure>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Step 3: Write the equation<\/strong><br>Now substitute <em>m<\/em> = \u22121 and <em>b<\/em> = 41 into the equation:<br><em>h<\/em>(<em>x<\/em>) = \u2212<em>x<\/em> + 41.<br><br><strong>Step 4: Verify the solution<\/strong><br>Substitute <em>x<\/em> = 0 and <em>x<\/em> = 1 into the equation to confirm it matches the given values:<br>~ For <em>x<\/em> = 0:<br><em>h<\/em>(0) = \u22120 + 41 = 41(correct).<br>~ For <em>x<\/em> = 1:<br><em>h<\/em>(1) = \u22121 + 41 = 40(correct).<br>Thus, the equation <em>h<\/em>(<em>x<\/em>) = \u2212<em>x<\/em> + 41 is correct.<br><br><strong>Final Answer:<\/strong> <strong>A) <em>h<\/em>(<em>x<\/em>) = \u2212<em>x<\/em> + 41<\/strong><\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>13th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> Which expression is equivalent to <strong>6x<sup>8<\/sup>y<sup>2<\/sup> + 12x<sup>2<\/sup>y<sup>2<\/sup><\/strong>?<br>A) 6x<sup>2<\/sup>y<sup>2<\/sup>(2x<sup>6<\/sup>)<br>B) 6x<sup>2<\/sup>y<sup>2<\/sup>(x<sup>4<\/sup>)<br>C) 6x<sup>2<\/sup>y<sup>2<\/sup>(x<sup>6<\/sup> + 2)<br>D) 6x<sup>2<\/sup>y<sup>2<\/sup>(x<sup>4<\/sup> + 2)<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Choice C<\/strong> is correct. Since each term of the given expression has a common factor of 6x<sup>2<\/sup>y<sup>2<\/sup>, it may be rewritten as 6x<sup>2<\/sup>y<sup>2<\/sup>(x<sup>6<\/sup>) + 6x<sup>2<\/sup>y<sup>2<\/sup>(2), or 6x<sup>2<\/sup>y<sup>2<\/sup>(x<sup>6<\/sup> + 2).<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice A is incorrect. This expression is equivalent to 12x<sup>8<\/sup>y<sup>2<\/sup>, not 6x<sup>8<\/sup>y<sup>2<\/sup> + 12x<sup>2<\/sup>y<sup>2<\/sup>.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice B is incorrect. This expression is equivalent to 6x<sup>6<\/sup>y<sup>2<\/sup>, not 6x<sup>8<\/sup>y<sup>2<\/sup> + 12x<sup>2<\/sup>y<sup>2<\/sup>.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice D is incorrect. This expression is equivalent to 6x<sup>6<\/sup>y<sup>2<\/sup>  + 12x<sup>2<\/sup>y<sup>2<\/sup>, not 6x<sup>8<\/sup>y<sup>2<\/sup> +12x<sup>2<\/sup>y<sup>2<\/sup>.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Step 1: Identify the common factors<\/strong><br>Both terms have the following factors in common:<br>~ The coefficient 6 (as both 6 and 12 are divisible by 6).<br>~ The variable x<sup>2<\/sup> (as the smallest power of <em>x<\/em> is x<sup>2<\/sup>).<br>~ The variable y<sup>2<\/sup> (as both terms include y<sup>2<\/sup>).<br>Thus, the greatest common factor (GCF) is 6x<sup>2<\/sup>y<sup>2<\/sup>.<br><br><strong>Step 2: Factor out the GCF<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"601\" height=\"640\" src=\"https:\/\/us.mrenglishkj.com\/sat\/sat\/wp-content\/uploads\/2024\/12\/image-3.png\" alt=\"Equivalent expressions of the SAT math problem solutions\" class=\"wp-image-4713\" srcset=\"https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/12\/image-3.png 601w, https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/12\/image-3-282x300.png 282w\" sizes=\"auto, (max-width: 601px) 100vw, 601px\" \/><\/figure>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Final Answer:<\/strong> <strong>C) 6x<sup>2<\/sup>y<sup>2<\/sup>(x<sup>6<\/sup> + 2)<\/strong><\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>14th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> A neighborhood consists of a 2-hectare park and a 35-hectare residential area. The total number of trees in the neighborhood is 3,934. The equation 2<em>x<\/em> + 35<em>y<\/em> = 3,934 represents this situation. Which of the following is the best interpretation of <em>x<\/em> in this context?<br>A) The average number of trees per hectare in the park<br>B) The average number of trees per hectare in the residential area<br>C) The total number of trees in the park<br>D) The total number of trees in the residential area<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Choice A<\/strong> is correct. It\u2019s given that a neighborhood consists of a 2-hectare park and a 35-hectare residential area and that the total number of trees in the neighborhood is 3,934. It\u2019s also given that the equation 2<em>x<\/em> + 35<em>y<\/em> = 3,934 represents this situation. Since the total number of trees for a given area can be determined by taking the number of hectares times the average number of trees per hectare, this must mean that the terms 2<em>x<\/em> and 35<em>y<\/em> correspond to the number of trees in the park and in the residential area, respectively. Since 2<em>x<\/em> corresponds to the number of trees in the park, and 2 is the size of the park, in hectares, <em>x<\/em> must represent the average number of trees per hectare in the park.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice B is incorrect and may result from conceptual errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice C is incorrect and may result from conceptual errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice D is incorrect and may result from conceptual errors.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Solution<\/strong><br>We are tasked with interpreting the variable <em>x<\/em> in the equation 2<em>x<\/em> + 35<em>y<\/em> = 3,934, which represents the total number of trees in a neighborhood. Let&#8217;s break this down step-by-step.<br><br><strong>Step 1: Understand the given equation<\/strong><br>The equation is: 2<em>x<\/em> + 35<em>y<\/em> = 3,934<br>This represents:<br>~ 2<em>x<\/em>: The number of trees in the 2-hectare park.<br>~ 35<em>y<\/em>: The number of trees in the 35-hectare residential area.<br>~ 3,934: The total number of trees in the neighborhood.<br>Thus, <em>x<\/em> and <em>y<\/em> are the <strong>average numbers of trees per hectare<\/strong> in the park and residential area, respectively.<br><br><strong>Step 2: Analyze <em>x<\/em><\/strong><br>From the term 2<em>x<\/em>, we see that <em>x<\/em> is multiplied by 2 (the size of the park in hectares). Therefore, <em>x<\/em> must represent the <strong>number of trees per hectare<\/strong> in the park.<br>Multiplying <em>x<\/em> by 2 gives the total number of trees in the 2-hectare park. Hence, <em>x<\/em> is not the <strong>total number of trees in the park<\/strong>, nor is it related to the residential area.<br><br><strong>Step 3: Eliminate incorrect options<\/strong><br>Let\u2019s analyze each option:<br>~ <strong>Option A: The average number of trees per hectare in the park<\/strong><br>This is <strong>correct<\/strong>, as <em>x<\/em> represents the number of trees per hectare in the 2-hectare park.<br>~ <strong>Option B: The average number of trees per hectare in the residential area<\/strong><br>This is <strong>incorrect<\/strong>, as <em>y<\/em>, not <em>x<\/em>, represents the number of trees per hectare in the residential area.<br>~ <strong>Option C: The total number of trees in the park<\/strong><br>This is <strong>incorrect<\/strong>, as 2<em>x<\/em>, not <em>x<\/em>, represents the total number of trees in the park.<br>~ <strong>Option D: The total number of trees in the residential area<\/strong><br>This is <strong>incorrect<\/strong>, as 35<em>y<\/em>, not <em>x<\/em>, represents the total number of trees in the residential area.<br><br><strong>Step 4: Verify the interpretation<\/strong><br>To confirm: <em>x<\/em> is the <strong>average number of trees per hectare in the park<\/strong>, which, when multiplied by 2 (the size of the park), gives the total number of trees in the park.<br><br><strong>Final Answer:<\/strong> <strong>A) The average number of trees per hectare in the park<\/strong><\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>15th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"270\" height=\"255\" src=\"https:\/\/us.mrenglishkj.com\/sat\/sat\/wp-content\/uploads\/2024\/12\/image-4.png\" alt=\"SAT Math problem solving and solutions with free explanation\" class=\"wp-image-4718\"\/><\/figure>\n\n\n\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> The graph shows the relationship between the number of shares of stock from Company A, <em>x<\/em>, and the number of shares of stock from Company B, <em>y<\/em>, that Simone can purchase. Which equation could represent this relationship?<br>A) <em>y<\/em> = 8<em>x<\/em> + 12<br>B) 8<em>x<\/em> + 12<em>y<\/em> = 480<br>C) y = 12<em>x<\/em> + 8<br>D) 12<em>x<\/em> + 8<em>y<\/em> = 480<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Choice B<\/strong> is correct. The graph shown is a line passing through the points (0, 40) and (60, 0). Since the relationship between <em>x<\/em> and <em>y<\/em> is linear, if two points on the graph make a linear equation true, then the equation represents the relationship. Substituting 0 for <em>x<\/em> and 40 for <em>y<\/em> in the equation in choice B,<br>8<em>x<\/em> +12<em>y<\/em> = 480, yields 8(0) + 12(40) = 480, or 480 = 480, which is true. Substituting 60 for <em>x<\/em> and 0 for <em>y<\/em> in the equation 8<em>x<\/em> +12<em>y<\/em> = 480 yields 8(60) + 12(0) = 480, or 480 = 480, which is true. Therefore, the equation 8<em>x<\/em> +12<em>y<\/em> = 480 represents the relationship between <em>x<\/em> and <em>y<\/em>.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice A is incorrect. The point (0, 40) is not on the graph of this equation, since 40 = 8(0) + 12, or 40 = 12, is not true.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice C is incorrect. The point (0, 40) is not on the graph of this equation, since 40 = 12(0) + 8, or 40 = 8, is not true.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice D is incorrect. The point (0, 40) is not on the graph of this equation, since 12(0) + 8(40) = 480, or 320 = 480, is not true.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Step-by-Step Solution:<\/strong><br><strong>Step 1: Understand the relationship<\/strong><br>The graph represents the relationship between the number of shares Simone can purchase from <strong>Company A (x)<\/strong> and <strong>Company B (y)<\/strong>. This relationship is linear, meaning it can be expressed as an equation of the form:<br><em>ax<\/em> + <em>by<\/em> = <em>c<\/em><br>where <em>a<\/em>, <em>b<\/em>, and <em>c<\/em> are constants determined by the context.<br><br><strong>Step 2: Interpret the graph&#8217;s key points<\/strong><br>From the graph:<br>~ When <em><strong>x<\/strong><\/em> = 0 (Simone buys no shares of Company A), <em><strong>y<\/strong><\/em> = 40 (she buys 40 shares of Company B).<br>~ When <strong><em>y<\/em><\/strong> = 0 (Simone buys no shares of Company B), <em><strong>x<\/strong><\/em> = 60 (she buys 60 shares of Company A).<br>These points are <strong>(0, 40)<\/strong> and <strong>(60, 0)<\/strong>, which are intercepts of the line.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"734\" height=\"536\" src=\"https:\/\/us.mrenglishkj.com\/sat\/sat\/wp-content\/uploads\/2024\/12\/image-5.png\" alt=\"Learn SAT Math by solving step-by-step solutions\" class=\"wp-image-4720\" srcset=\"https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/12\/image-5.png 734w, https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/12\/image-5-300x219.png 300w\" sizes=\"auto, (max-width: 734px) 100vw, 734px\" \/><\/figure>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Step 4: Verify the equation<\/strong><br>To confirm, substitute the intercepts: <em>ax<\/em> + <em>by<\/em> = <em>c<\/em><br>where <em>a<\/em> = 8, <em>b<\/em> = 12, <em>c<\/em> = 480, and (<em>x<\/em>, y) = (0, 40) or (60, 0).<br>1. For <em>x<\/em> = 0:<br>8(0) + 12y = 480<br>\u200a\u27f9\u2005\u200a12<em>y<\/em> = 480<br>\u200a\u27f9\u2005\u200ay = 480\/12<br>\u200a\u27f9\u2005<em>\u200ay<\/em> = 40<br>This matches the graph (<em>x, y<\/em>) = (0, 40). If <em>x<\/em> = 0 then <em>y<\/em> = 40.<br>2. For <em>y<\/em> = 0:<br>8<em>x<\/em> + 12(0) = 480<br>\u27f9\u2005\u200a8<em>x<\/em> = 480<br>\u27f9\u2005\u200ax = 480\/8<br>\u27f9\u2005\u200a<em>x<\/em> = 60<br>This also matches the graph (<em>x, y<\/em>) = (60, 0). If <em>y<\/em> = 0 then <em>x<\/em> = 60.<br>Thus, the equation 8<em>x<\/em> + 12<em>y<\/em> = 480 correctly represents the relationship.<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>16th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> The regular price of a bag at a store is $60. The sale price of the bag is 50% less than the regular price, and the sale price is 20% greater than the store\u2019s cost for the bag. What was the store\u2019s cost, in dollars, for the bag?<br>A) $20<br>B) $24<br>C) $25<br>D) $30<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Choice C<\/strong> is the correct answer.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice A is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice B is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice D is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Expanded Step-by-Step Solution<\/strong><br><strong>Step 1: Understanding the problem<\/strong><br>The question describes a relationship between the regular price, the sale price, and the store\u2019s cost:<br>1. The regular price of the bag is <strong>$60<\/strong>.<br>2. The sale price is <strong>50% less<\/strong> than the regular price.<br>3. The sale price is <strong>20% greater<\/strong> than the store\u2019s cost.<br>The goal is to calculate the <strong>store\u2019s cost<\/strong> based on these relationships.<br><br><strong>Step 2: Calculating the sale price<\/strong><br>The sale price is <strong>50% less than the regular price<\/strong>. To find this:<br>1. Start by calculating 50% of the regular price. Recall that <strong>50%<\/strong> means half of the value or multiplying the value by <strong>0.5<\/strong>.<br>50% \u00d7 60<br>0.5 \u00d7 60 = 30<br>2. Subtract this value from the regular price:<br>Sale&nbsp;Price = Regular&nbsp;Price \u2212 (50% \u00d7 Regular&nbsp;Price)<br>Sale&nbsp;Price = 60 \u2212 30 = 30<br>Thus, the <strong>sale price is $30<\/strong>.<br><br><strong>Step 3: Relating the sale price to the store&#8217;s cost<\/strong><br>The problem states that the sale price is <strong>20% greater than the store\u2019s cost<\/strong>. To express this mathematically:<br>1. Let the store&#8217;s cost be <em>x<\/em>.<br>2. The sale price is 20% more than <em>x<\/em>. Add 20% of <em>x<\/em> to <em>x<\/em>:<br>Sale&nbsp;Price = <em>x<\/em> + 0.2<em>x<\/em><br>Combine like terms:<br>Sale&nbsp;Price = 1.2<em>x<\/em><br>The equation becomes:<br>30 = 1.2<em>x<\/em><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"684\" height=\"602\" src=\"https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/12\/Screenshot-2024-12-01-191320.png\" alt=\"SAT math problem solutions step-by-step\" class=\"wp-image-4660\" srcset=\"https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/12\/Screenshot-2024-12-01-191320.png 684w, https:\/\/us.mrenglishkj.com\/sat\/wp-content\/uploads\/sites\/2\/2024\/12\/Screenshot-2024-12-01-191320-300x264.png 300w\" sizes=\"auto, (max-width: 684px) 100vw, 684px\" \/><\/figure>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Step 5: Verifying the solution<\/strong><br>Let\u2019s confirm the calculation:<br>1. If the store&#8217;s cost is <strong>$25<\/strong>, then 20% of $25 is:<br>20% \u00d7 25<br>0.2 \u00d7 25 = 5<br>2. Add this value to the store&#8217;s cost:<br>25 + 5 = 30<br>The sale price is $30, matching the earlier calculation. This confirms the solution.<br><br><strong>Final Answer:<\/strong> <strong>C) $25<\/strong><\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>17th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question: <\/strong>Kaylani used fabric measuring 5 yards in length to make each suit for a men\u2019s choir. The relationship between the number of suits that Kaylani made, <em>x<\/em>, and the total length of fabric that she purchased <em>y<\/em>, in yards, is represented by the equation <em>y<\/em> \u2212 5<em>x<\/em> = 6. What is the best interpretation of 6 in this context?<br>A) Kaylani made 6 suits.<br>B) Kaylani purchased a total of 6 yards of fabric.<br>C) Kaylani used a total of 6 yards of fabric to make the suits.<br>D) Kaylani purchased 6 yards more fabric than she used to make the suits.<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Choice D<\/strong> is correct. It\u2019s given that the equation <em>y<\/em> &#8211; 5<em>x<\/em> = 6 represents the relationship between the number of suits that Kaylani made, <em>x<\/em>, and the total length of fabric she purchased, <em>y<\/em>, in yards. Adding 5<em>x<\/em> to both sides of the given equation yields <em>y<\/em> = 5<em>x<\/em> + 6. Since Kaylani made <em>x<\/em> suits and used 5 yards of fabric to make each suit, the expression 5<em>x<\/em> represents the total amount of fabric she used to make the suits. Since <em>y<\/em> represents the total length of fabric Kaylani purchased, in yards, it follows from the equation <em>y<\/em> = 5<em>x<\/em> + 6 that Kaylani purchased 5<em>x<\/em> yards of fabric to make the suits, plus an additional 6 yards of fabric. Therefore, the best interpretation of 6 in this context is that Kaylani purchased 6 yards more fabric than she used to make the suits.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice A is incorrect. Kaylani made a total of <em>x<\/em> suits, not 6 suits.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice B is incorrect. Kaylani purchased a total of <em>y<\/em> yards of fabric, not a total of 6 yards of fabric.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice C is incorrect. Kaylani used a total of 5<em>x<\/em> yards of fabric to make the suits, not a total of 6 yards of fabric.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Problem Analysis<\/strong><br>The given equation is: <em>y<\/em> \u2212 5<em>x<\/em> = 6<br>Here:<br>~ <em>y<\/em> represents the total length of fabric Kaylani purchased (in yards).<br>~ <em>x<\/em> represents the number of suits Kaylani made.<br>~ Each suit requires 5 yards of fabric.<br>We need to interpret the constant 6 in the equation.<br><br><strong>Step-by-Step Solution<\/strong><br><strong>Step 1: Rearrange the equation<\/strong><br>Rearrange the given equation to isolate <em>y<\/em> (the total fabric purchased): <em>y<\/em> = 5<em>x<\/em> + 6<br>From this form, we see:<br>~ 5<em>x<\/em> represents the total fabric required to make <em>x<\/em> suits, where each suit uses 5 yards.<br>~ 6 is an additional amount of fabric <strong>purchased beyond the amount needed for making the suits.<\/strong><br><br><strong>Step 2: Interpret 6<\/strong><br>The constant 6 indicates that Kaylani purchased 6 more yards of fabric than the amount required to make the suits. In other words, it represents <strong>extra fabric purchased<\/strong>.<br><br><strong>Step 3: Verify with an example<\/strong><br>Let\u2019s verify the interpretation by substituting values for <em>x<\/em>.<br>1. If Kaylani makes <em>x<\/em> = 0 suits, the equation becomes:<br><em>y<\/em> = 5(0) + 6<br><em>y<\/em> = 6<br>This means she purchased 6 yards of fabric even when no suits were made.<br>2. If Kaylani makes <em>x<\/em> = 2 suits, the equation becomes:<br><em>y<\/em> = 5(2) + 6<br><em>y<\/em> =10 + 6<br><em>y<\/em> =16<br>This means she purchased 16 yards of fabric, 10 of which were used for the suits, and 6 yards were extra.<br>Thus, 6 is the <strong>extra fabric purchased<\/strong>.<br><br><strong>Step 4: Identify the correct option<\/strong><br>From the given options:<br>~ <strong>A)<\/strong> &#8220;Kaylani made 6 suits&#8221; is incorrect because 6 is not related to the number of suits made.<br>~ <strong>B)<\/strong> &#8220;Kaylani purchased a total of 6 yards of fabric&#8221; is incorrect because 6 is the extra fabric, not the total fabric.<br>~ <strong>C)<\/strong> &#8220;Kaylani used a total of 6 yards of fabric to make the suits&#8221; is incorrect because the fabric used for suits is 5<em>x<\/em>, not 6.<br>~ <strong>D)<\/strong> &#8220;Kaylani purchased 6 yards more fabric than she used to make the suits&#8221; is correct because 6 represents the extra fabric purchased.<br><strong>The correct answer is: Option D<\/strong>\u200b<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>18th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> For <em>x<\/em> &gt; 0, the function <em>g<\/em> is defined as follows: <em>g<\/em>(<em>x<\/em>) equals <strong>125% of <em>x<\/em><\/strong>. Which of the following could describe this function?<br>A) Decreasing exponential<br>B) Decreasing linear<br>C) Increasing exponential<br>D) Increasing linear<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Choice D<\/strong> is the correct answer.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice A is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice B is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice C is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Step-by-Step Solution<\/strong>:<br><strong>Step 1: Understanding the function definition<\/strong><br>The function <em>g<\/em>(<em>x<\/em>) is defined as <strong>125% of <em>x<\/em><\/strong>.<br>This means: <em>g<\/em>(<em>x<\/em>) = 125% multiplying by <em>x<\/em><br>To convert the percentage into a decimal form:<br>125% = 125\/100<br>125% = 1.25<br>Thus, the function becomes: <em>g<\/em>(<em>x<\/em>) = 1.25<em>x<\/em><br>This represents a linear relationship because <em>g<\/em>(<em>x<\/em>) is proportional to <em>x<\/em>, with the constant multiplier 1.25.<br><br><strong>Step 2: Identifying the function type<\/strong><br>1. <strong>Linear nature:<\/strong><br>The equation <em>g<\/em>(<em>x<\/em>) = 1.25<em>x<\/em> follows the general form of a linear function:<br><em>y<\/em> = <em>mx<\/em> + <em>b<\/em><br>Here, <em>m<\/em> = 1.25 (the slope), and <em>b<\/em> = 0 (no <em>y<\/em>-intercept other than the origin).<br>2. <strong>Increasing behavior:<\/strong><br>Since <em>m<\/em> = 1.25 &gt; 0, the function increases as <em>x<\/em> increases.<br>3. <strong>Not exponential:<\/strong><br>An exponential function has the form <em>y<\/em> = <em>ab<\/em><sup>x<\/sup>, where b\u22601. The function <em>g<\/em>(<em>x<\/em>) = 1.25<em>x<\/em> does not fit this form.<br><br><strong>Step 3: Eliminating incorrect options<\/strong><br>Let\u2019s analyze each option:<br><strong>A) Decreasing exponential:<\/strong> Incorrect. The function is not exponential and does not decrease.<br><strong>B) Decreasing linear:<\/strong> Incorrect. Although the function is linear, it is increasing because <em>m<\/em> &gt; 0.<br><strong>C) Increasing exponential:<\/strong> Incorrect. While the function increases, it is not exponential.<br><strong>D) Increasing linear:<\/strong> Correct. The function increases linearly with a constant slope of 1.25.<br><br><strong>Step 4: Verification<\/strong><br>For <em>x<\/em> &gt; 0, substitute sample values of <em>x<\/em> to confirm the behavior:<br>~ If <em>x<\/em> = 1, then <em>g<\/em>(<em>x<\/em>) = 1.25(1) = 1.25.<br>~ If <em>x<\/em> = 2, then <em>g<\/em>(<em>x<\/em>) = 1.25(2) = 2.5.<br>~ If <em>x<\/em> = 3, then <em>g<\/em>(<em>x<\/em>) = 1.25(3) = 3.75.<br>As <em>x<\/em> increases, <em>g<\/em>(<em>x<\/em>) increases proportionally, confirming the linear nature of the function.<br><br><strong>Final Answer:<\/strong> <strong>D) Increasing linear<\/strong><\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>19th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> A rectangular park has an area of <strong>3.25 square miles<\/strong>. What is the area, in square yards, of this park? (1 mile = 1,760 yards)<br>A) 8,540<br>B) 620,480<br>C) 5,004,800<br>D) 9,984,400<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Choice C<\/strong> is the correct answer.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice A is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice B is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice D is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Step-by-Step Solution<\/strong><br><strong>Step 1: Understanding the problem<\/strong><br>We are tasked with converting an area from <strong>square miles<\/strong> to <strong>square yards<\/strong>. The given area is 3.25\u2009square&nbsp;miles.<br>We know:<br>1\u2009mile = 1,760\u2009yards.<br>To find the area in square yards, we first convert the unit of miles to yards and then square the result.<br><br><strong>Step 2: Converting miles to yards<\/strong><br>Since 1\u2009mile = 1,760\u2009yards, the length of one side of a square mile is:<br>1\u2009mile = 1,760\u2009yards<br>To find the area in square yards, square this conversion factor:<br>(1\u2009mile)<sup>2<\/sup> = (1,760\u2009yards)<sup>2<\/sup><br>(1\u2009mile)<sup>2<\/sup> = 1,760 \u00d7 1,760 = 3,097,600\u2009square&nbsp;yards.<br>Thus, 1\u2009square&nbsp;mile = 3,097,600\u2009square&nbsp;yards.<br><br><strong>Step 3: Multiply by the given area in square miles<\/strong><br>The park has an area of 3.25\u2009square&nbsp;miles.<br>To find the area in square yards, multiply 3.25 by 3,097,600:<br>3.25 \u00d7 3,097,600 = 10,466,200\u2009square&nbsp;yards.<br><br><strong>Step 4: Verification of Calculation<\/strong><br>Break down the multiplication for verification:<br>3.25 \u00d7 3,097,600 = (3 \u00d7 3,097,600) + (0.25 \u00d7 3,097,600).<br>~ Compute 3 \u00d7 3,097,600:<br>3 \u00d7 3,097,600 = 9,292,800.<br>~ Compute 0.25 \u00d7 3,097,6000:<br>0.25 \u00d7 3,097,600 = 774,400.<br>~ Add the two results:<br>9,292,800 + 774,400 = 10,466,200.<br>Thus, the total area is correctly calculated as <strong>10,466,200<\/strong>\u2009square&nbsp;yards.<\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>20th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> In the xy-plane, a parabola has vertex (6, \u221210) and intersects the <em>x<\/em>-axis at two points. If the equation of the parabola is written in the form <em>y<\/em> = <em>ax<\/em><sup>2<\/sup> + <em>bx<\/em> + <em>c<\/em>, where <em>a<\/em>, <em>b<\/em>, and <em>c<\/em> are constants, which of the following could be the value of <em>a<\/em> + <em>b<\/em> + <em>c<\/em>?<br>A) \u221215<br>B) 15<br>C) 37<br>D) -37<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Choice B<\/strong> is correct.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice A is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice C is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice D is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Step-by-Step Solution<\/strong><br><strong>Step 1: Understand the Vertex Form of the Parabola<\/strong><br>The equation of a parabola in vertex form is:<br><em>y<\/em> = <em>a<\/em>(<em>x<\/em> \u2212 <em>h<\/em>)<sup>2<\/sup> + k,<br>where (<em>h<\/em>, <em>k<\/em>) is the vertex.<br>For this parabola, the vertex is given as (6, \u221210), so:<br><em>y<\/em> = <em>a<\/em>(<em>x<\/em> \u2212 6)<sup>2<\/sup> \u2212 10.<br><br><strong>Step 2: Determine the Standard Form<\/strong><br>Expand <em>y<\/em> = <em>a<\/em>(<em>x<\/em> \u2212 6)<sup>2<\/sup> \u2212 10 to convert it to standard form:<br>It means (x &#8211; 6)<sup>2<\/sup> gives algebric formula (<em>a<\/em> &#8211; <em>b<\/em>)<sup>2<\/sup> = (<em>a<\/em><sup>2<\/sup> &#8211; 2<em>ab<\/em> + b<sup>2<\/sup>). Hence: <em>y<\/em> = <em>a<\/em>(<em>x<\/em><sup>2<\/sup> \u2212 12<em>x<\/em> + 36) \u2212 10.<br>Distribute <em>a<\/em>:<br><em>y<\/em> = <em>ax<\/em><sup>2<\/sup> \u2212 12<em>ax<\/em> + 36<em>a<\/em> \u2212 10.<br>This gives the standard form:<br><em>y<\/em> = <em>ax<\/em><sup>2<\/sup> \u2212 12<em>ax<\/em> + (36<em>a<\/em> \u2212 10).<br>From the standard form, the coefficients are:<br>~ <em>a<\/em> = <em>a<\/em>,<br>~ <em>b<\/em> = \u221212<em>a<\/em>,<br>~ <em>c<\/em> = 36<em>a<\/em> \u2212 10.<br><br><strong>Step 3: Use the Condition for Intersection with the <em>x<\/em>-Axis<\/strong><br>For the parabola to intersect the <em>x<\/em>-axis at two points, the discriminant of the quadratic equation must be positive (\u0394 &gt; 0):<br>\u0394 = <em>b<\/em><sup>2<\/sup> \u2212 4<em>ac<\/em>.<br>Substitute <em>b<\/em> = \u221212<em>a<\/em> and <em>c<\/em> = 36<em>a<\/em> \u2212 10 into the discriminant:<br>\u0394 = (\u221212<em>a<\/em>)<sup>2<\/sup> \u22124(<em>a<\/em>)(36<em>a<\/em> \u2212 10).<br>Simplify: \u0394 = 144<em>a<\/em><sup>2<\/sup> \u2212 4(36<em>a<\/em><sup>2<\/sup> \u2212 10<em>a<\/em>).<br>\u0394 = 144<em>a<\/em><sup>2<\/sup> \u2212 144<em>a<\/em><sup>2<\/sup> + 40<em>a<\/em>.<br>\u0394 = 40<em>a<\/em>.<br>Since \u0394 &gt; 0, <em>a<\/em> &gt; 0.<br><br><strong>Step 4: Solve for <em>a<\/em> + <em>b<\/em> + <em>c<\/em><\/strong><br>Choose a simple value for <em>a<\/em> (e.g., <em>a<\/em> = 1, as <em>a<\/em> &gt; 0):<br>~ <em>b<\/em> = \u221212<em>a<\/em><br><em>b<\/em> = \u221212(1)<br><em>b<\/em> = \u221212<br>~ <em>c<\/em> = 36<em>a<\/em> \u2212 10<br><em>c<\/em> = 36(1) \u2212 10<br><em>c<\/em> = 36 &#8211; 10<br><em>c<\/em> = 26.<br>Now calculate <em>a<\/em> + <em>b<\/em> + <em>c<\/em>:<br><em>a<\/em> + <em>b<\/em> + <em>c<\/em><br>1 \u2212 12 + 26 =15.<br>Thus, <em>a<\/em> + <em>b<\/em> + <em>c<\/em> = 15.<br><br><strong>Verification: <\/strong><br>Substitute back into the equation and check if the parabola still satisfies the conditions (e.g., correct vertex and intersects the <em>x<\/em>-axis):<br>~ Equation: y = (x \u2212 6)2 \u2212 10, expanded as y = x2 \u2212 12x + 26.<br>~ Vertex is (6, \u221210), as required, and discriminant \u0394 &gt; 0, confirming intersection with the <em>x<\/em>-axis.<br><br><strong>Final Answer: B) 15<\/strong><\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>21th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> The number <em>p<\/em> is <strong>85% greater<\/strong> than the number <em>q<\/em>. The number <em>q<\/em> is <strong>75% less<\/strong> than 60. What is the value of <em>p<\/em>?<br>A) 15<br>B) 15.75<br>C) 25.25<br>D) 27.75<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Choice D<\/strong> is the correct answer.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice A is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice B is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice C is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Step-by-Step Solution<\/strong><br><strong>Step 1: Calculate the Value of <em>q<\/em><\/strong><br>The problem states that <em>q<\/em> is <strong>75% less than 60<\/strong>. To find <em>q<\/em>:<br><em>q<\/em> = 60 \u2212 (75%\u2009 of 60)<br><em>q<\/em> = 60 \u2212 0.75 \u00d7 60<br><em>q<\/em> = 60 \u2212 45<br>Thus, <em>q<\/em> = 15.<br><br><strong>Step 2: Calculate the Value of <em>p<\/em><\/strong><br>The problem states that <em>p<\/em> is <strong>85% greater than <em>q<\/em><\/strong>. This means <em>p<\/em> is equal to <em>q<\/em> plus 85% of <em>q<\/em>. To find <em>p<\/em>:<br><em>p<\/em> = <em>q<\/em> + (85%\u2009of&nbsp;<em>q<\/em>)<br><em>p<\/em> = <em>q<\/em> + 0.85 \u00d7 <em>q<\/em><br><em>p<\/em> = 15 + 0.85 \u00d7 15<br><em>p<\/em> = 15 + 12.75 = 27.75<br>Thus, <em>p<\/em> = 27.75.<br><br><strong>Verification<\/strong><br>~ Calculated q = 15 as 75% less than 60.<br>~ Calculated <em>p<\/em> = 27.75 as 85% greater than <em>q<\/em>.<br>Both steps are consistent with the problem&#8217;s conditions.<br><br><strong>Final Answer: D) 27.75<\/strong><\/p>\n<\/div><\/details><\/div>\n\n\n\n<div class=\"wp-block-coblocks-accordion-item\"><details><summary class=\"wp-block-coblocks-accordion-item__title\"><strong>22th Question<\/strong><\/summary><div class=\"wp-block-coblocks-accordion-item__content\">\n<p class=\"is-style-warning\" style=\"font-size:0.9em\"><strong>Question:<\/strong> The function <em>g<\/em> is defined by <em>g<\/em>(<em>x<\/em>) = <em>x<\/em>(<em>x<\/em> \u2212 2)(<em>x<\/em> + 6)<sup>2<\/sup>. The value of <em>g<\/em>(7\u2212  <em>w<\/em>)is 0, where <em>w<\/em> is a constant. What is the sum of all possible values of <em>w<\/em>?<br>A) 25<br>B) 7<br>C) 5<br>D) 13<br>[Type-Based Answer: In the final exam, you will type the answer rather than choose from options.]<\/p>\n\n\n\n<p class=\"is-style-success\" style=\"font-size:0.9em\"><strong>Option A:<\/strong> The correct answer is 25. The value of <em>g<\/em>(7 &#8211; <em>w<\/em>) is the value of <em>g<\/em>(<em>x<\/em>) when <em>x<\/em> = 7 &#8211; <em>w<\/em>, where <em>w<\/em> is a constant. Substituting 7 &#8211; <em>w<\/em> for <em>x<\/em> in the given equation yields <em>g<\/em>(7 &#8211; <em>w<\/em>) = (7 &#8211; <em>w<\/em>)(7 &#8211; <em>w<\/em> &#8211; 2)(7 &#8211; <em>w<\/em> + 6)<sup>2<\/sup>, which is equivalent to <em>g<\/em>(7 &#8211; <em>w<\/em>) = (7 &#8211; <em>w<\/em>)(5 &#8211; <em>w<\/em>)(13 &#8211; w)<sup>2<\/sup>. It\u2019s given that the value of <em>g<\/em>(7 &#8211; <em>w<\/em>) is 0. Substituting 0 for <em>g<\/em>(7 &#8211; <em>w<\/em>) in the equation <em>g<\/em>(7 &#8211; <em>w<\/em>) = (7 &#8211; <em>w<\/em>)(5 &#8211; <em>w<\/em>)(13 &#8211; <em>w<\/em>)<sup>2<\/sup> yields 0 = (7 &#8211; <em>w<\/em>)(5 &#8211; <em>w<\/em>)(13 &#8211; w)<sup>2<\/sup>. Since the product of the three factors on the right-hand side of this equation is equal to 0, at least one of these three factors must be equal to 0. Therefore, the possible values of <em>w<\/em> can be found by setting each factor equal to 0. Setting the first factor equal to 0 yields 7 &#8211; <em>w<\/em> = 0. Adding <em>w<\/em> to both sides of this equation yields 7 = <em>w<\/em>. Therefore, 7 is one possible value of <em>w<\/em>. Setting the second factor equal to 0 yields 5 &#8211; <em>w<\/em> = 0. Adding <em>w<\/em> to both sides of this equation yields 5 = <em>w<\/em>. Therefore, 5 is a second possible value of <em>w<\/em>. Setting the third factor equal to 0 yields (13 &#8211; <em>w)<\/em><sup>2<\/sup> = 0. Taking the square root of both sides of this equation yields 13 &#8211; w = 0. Adding <em>w<\/em> to both sides of this equation yields 13 = <em>w<\/em>. Therefore, 13 is a third possible value of <em>w<\/em>. Adding the three possible values of <em>w<\/em> yields 7 + 5 + 13, or 25. Therefore, the sum of all possible values of <em>w<\/em> is 25.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice B is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice C is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-error\" style=\"font-size:0.9em\">Choice D is incorrect and may result from conceptual or calculation errors.<\/p>\n\n\n\n<p class=\"is-style-info\" style=\"font-size:0.9em\"><strong>Problem Analysis:<\/strong><br>The given function is:<br><em>g<\/em>(<em>x<\/em>) = <em>x<\/em>(<em>x<\/em> \u2212 2)(<em>x<\/em> + 6)<sup>2<\/sup><br>The question states that <em>g<\/em>(7 \u2212 <em>w<\/em>) = 0, and we are to find the <strong>sum of all possible values of <em>w<\/em><\/strong>.<br>For <em>g<\/em>(7 \u2212 <em>w<\/em>) to equal 0, any of the factors of <em>g<\/em>(<em>x<\/em>) must be 0 because a product is 0 if at least one of its factors is 0.<br><br><strong>Step-by-Step Solution:<\/strong><br><strong>Step 1: Replace <em>x<\/em> with 7 \u2212 <em>w<\/em><\/strong><br>Substitute 7 \u2212 <em>w<\/em> for <em>x<\/em> in <em>g<\/em>(<em>x<\/em>): <em>x<\/em>(<em>x<\/em> \u2212 2)(<em>x<\/em> + 6)<sup>2<\/sup><br><em>g<\/em>(7 \u2212 <em>w<\/em>) = (7 \u2212 <em>w<\/em>)[(7 \u2212 <em>w<\/em>) \u2212 2][(7 \u2212 <em>w<\/em>) + 6]<sup>2<\/sup><br>Simplify each term:<br>~ The first factor: 7 \u2212 <em>w<\/em><br>~ The second factor: (7 \u2212 <em>w<\/em>) \u2212 2<br>7 &#8211; 2 &#8211; <em>w<\/em><br>5 \u2212 <em>w<\/em><br>~ The third factor: (7 \u2212 <em>w<\/em>) + 6<br>7 + 6 &#8211; <em>w<\/em><br>13 \u2212 <em>w<\/em><br>So:<br><em>g<\/em>(7 \u2212 <em>w<\/em>) = (7 \u2212 <em>w<\/em>)(5 \u2212 <em>w<\/em>)(13 \u2212 <em>w<\/em>)<sup>2<\/sup><br><br><strong>Step 2: Solve for <em>g<\/em>(7 \u2212 <em>w<\/em>) = 0<\/strong><br>For <em>g<\/em>(7 \u2212 <em>w<\/em>) = 0, at least one of the factors must equal 0. Solve each factor:<br>~ First factor: 7 \u2212 <em>w<\/em> = 0  \u27f9\u2005\u200a<em>w<\/em> = 7<br>~ Second factor: 5 \u2212 <em>w<\/em> = 0\u2005\u200a\u27f9\u2005\u200a<em>w<\/em> = 5<br>~ Third factor: 13 \u2212 <em>w<\/em> = 0\u2005\u200a\u27f9\u2005\u200a<em>w<\/em> = 13<br><br><strong>Step 3: Find the sum of all possible values of <em>w<\/em><\/strong><br>~ The possible values of <em>w<\/em> are:<br><em>w<\/em> = 7,\u2009<em>w<\/em> = 5,\u2009<em>w<\/em> = 13<br>~ The sum of these values is: 7 + 5 + 13 = 25<br><br><strong>Final Answer:<\/strong><br>The sum of all possible values of <em>w<\/em> is: <strong>A) 25<\/strong>\u200b<\/p>\n<\/div><\/details><\/div>\n<\/div>\n\n\n\n<div style=\"height:70px\" aria-hidden=\"true\" class=\"wp-block-spacer\"><\/div>\n\n\n\n<p>Did you try all the features and get comfortable using them? You should work on using the calculator and seeing references and directions. So be prepared for everything before taking the final SAT exam. The explanation of answers makes it easy to learn and progress. You must try to work on your speed and spend less time on the beginning and more on the later questions. This is the 4th Practice Test of SAT Math Module 1st.<\/p>\n\n\n\n<p>Either you can take the 5th Practice Test of SAT Math or the 4th Practice Test of SAT Math Module 2nd.<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li><a href=\"https:\/\/us.mrenglishkj.com\/sat\/sat-math-test-4-module-2nd-preparation\/\" target=\"_blank\" rel=\"noopener\" title=\"\">SAT Test 4th (Math Module 2nd)<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/us.mrenglishkj.com\/sat\/sat-math-test-5-module-1st-prepare-learn\/\" target=\"_blank\" rel=\"noopener\" title=\"\">SAT Test 5th (Math Module 1st)<\/a><\/li>\n\n\n\n<li><a href=\"https:\/\/us.mrenglishkj.com\/sat\/sat-qa-4-reading-and-writing-1st-module\/\" target=\"_blank\" rel=\"noopener\" title=\"\">SAT Test 4th (Reading and Writing Module 1st)<\/a><\/li>\n<\/ul>\n\n\n\n<p>The best way to become a master in Math is to find the correct answer and understand why other options are incorrect. I wish you luck in your bright career.<\/p>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>SAT Math Test 4 Module 1st with Simple Solution and Hack: Prepare yourself to take the SAT Math Test for free by taking our real-time SATs, the best part is there are solutions and step-by-step explanations with tricks<\/p>\n","protected":false},"author":1,"featured_media":4624,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"googlesitekit_rrm_CAowmvTFDA:productID":"","_coblocks_attr":"","_coblocks_dimensions":"","_coblocks_responsive_height":"","_coblocks_accordion_ie_support":"","footnotes":""},"categories":[12],"tags":[27,28],"class_list":["post-4621","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-1st-module","tag-sat-math","tag-sat-module-1st"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/posts\/4621","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/comments?post=4621"}],"version-history":[{"count":1,"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/posts\/4621\/revisions"}],"predecessor-version":[{"id":8824,"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/posts\/4621\/revisions\/8824"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/media\/4624"}],"wp:attachment":[{"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/media?parent=4621"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/categories?post=4621"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/us.mrenglishkj.com\/sat\/wp-json\/wp\/v2\/tags?post=4621"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}