Question: What is the y-intercept of the graph shown?
A) (−8, 0)
B) (−6, 0)
C) (0, 6)
D) (0, 8)

Choice D 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).

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Step 1: Understanding the y-intercept
The y-intercept is the point where the graph intersects the y-axis.
~ On the y-axis, x = 0, so the y-intercept is the value of y when x = 0.
~ This means we are looking for the coordinate of the form (0, y) on the graph.

Step 2: Observing the graph at x = 0
~ From the graph provided, when x = 0, the value of y is clearly at 8.
~ The corresponding point is (0, 8).

Step 3: Choosing the correct option
~ The y-intercept is (0, 8).
~ Among the given options:
~ ~ Option A: (−8, 0) = Incorrect; this is not on the y-axis.
~ ~ Option B: (−6, 0) = Incorrect; this is also not on the y-axis.
~ ~ Option C: (0, 6) = Incorrect; the graph does not pass through y = 6 when x = 0.
~ ~ Option D: (0, 8) = Correct; this matches the point where the graph intersects the y-axis.
Thus, Option D is the correct answer.

Question: The total cost f(x), in dollars, to lease a car for 36 months from a particular car dealership is given by f(x) = 36x + 1,000, where x is the monthly payment, in dollars. What is the total cost to lease a car when the monthly payment is dollar sign $400?
A) $13,400
B) $13,000
C) $15,400
D) $37,400

Choice C is correct. It’s 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.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Step 1: Understand the function and given values
The function is provided as:
f(x) = 36x + 1,000
~ f(x) represents the total cost of leasing the car for 36 months, in dollars.
~ x represents the monthly payment, in dollars.
~ We are asked to find f(x) when x = 400 (monthly payment is $400).

Step 2: Substitute x = 400 into the function
To calculate the total cost f(x) when x = 400, substitute 400 for x in the function:
f(x) = 36x + 1,000
f(400) = 36(400) +1,000

Step 3: Perform the calculations
~ Multiply 36 × 400:
36 × 400 = 14,400
~ Add 1,000 to the result:
f(400) = 14,400 + 1,000 = 15,400

Step 4: State the final answer
The total cost to lease the car when the monthly payment is $400 is: 15,400 dollars.​

Question: Each side of a square has a length of 45. What is the perimeter of this square?
A) 45
B) 180
C) 90
D) 360
[Type-Based Answer: In the final exam, you will type the answer rather than choose from options.]

Option B: 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’s 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.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Step 1: Understand the problem
The question gives:
~ The length of each side of a square = 45.
~ We need to find the perimeter of the square.
The formula for the perimeter of a square is:
Perimeter= 4 × side length

Step 2: Substitute the side length
Substitute 45 for the side length in the formula: Perimeter= 4 × 45

Step 3: Perform the calculation
4 × 45 = 180

Step 4: State the final answer
The perimeter of the square is: 180 units​

Question:

A) 11
B) -11
C) 55
D) 5
[Type-Based Answer: In the final exam, you will type the answer rather than choose from options.]

Option D: 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 = x² + 6x. Subtracting 55 from both sides of this equation results in 0 = x² + 6x – 55. The right-hand side of this equation can be rewritten by factoring. The two values that multiply to
-55 and add to 6 are 11 and -5. It follows that the equation 0 = x² + 6x – 55 can be rewritten as 0 = (x + 11)(x – 5). Setting each factor equal to 0 yields two equations: x + 11 = 0 and x – 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 – 5 = 0 results in x = 5. Therefore, the positive solution to the given equation is 5.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

We are tasked with finding the positive solution to this equation.
Step 1: Eliminate the fraction
Multiply both sides of the equation by x + 6 (the denominator) to eliminate the fraction. This step ensures that we don’t divide by zero later.
55 = x(x + 6)

Step 2: Expand the equation
Expand the right-hand side: 55 = x² + 6x

Step 3: Rearrange into standard quadratic form
Move all terms to one side of the equation to set it equal to 0:
x² + 6x − 55 = 0

Step 4: Solve the quadratic equation
The equation x² + 6x − 55 = 0 is a standard quadratic equation. We will solve it using the quadratic formula:

Step 5: Find the two solutions
Now, calculate the two possible values of x:

The solution is verified.

Final Answer
The positive solution is: 5​

Question: 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?
A) 9
B) 96
C) 120
D) 972

Choice A 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.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Step-by-Step Solution
Step 1: Understand the relationship between speed, distance, and time

Step 3: Perform the calculation
Divide 108 by 12:
Time = 9 seconds

Step 4: Verify
At a speed of 12 cm/s, traveling for 9 seconds would result in:
Distance = Speed × Time
Distance = 12 × 9
Distance = 108 cm
The calculation checks out.

Final Answer: 9 seconds.

Question: Right triangles △ABC and △DEF are similar, where A and B correspond to D and E, respectively. If ∠B measures 45^∘, what is the measure of ∠E?
A) 35^∘
B) 45^∘
C) 55^∘
D) 90^∘

Choice B is correct. Since we know both △ABC and △DEF are similar and if ∠B measures 45^∘ then you should focus on A = D, B = E, and C = F. Hence ∠E is 45^∘.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Step 1: Understand triangle similarity properties
Since the two triangles, △ABC and △DEF are similar, their corresponding angles are congruent. This means:
∠B = ∠E
Thus, if ∠B measures 45^∘, then ∠E must also measure 45^∘.

Step 2: Verify other triangle angle properties
In a triangle, the sum of all interior angles is always 180^∘. For both triangles △ABC and △DEF, this holds true. Therefore:
~ In △ABC, the angles are ∠A + ∠B + ∠C = 180^∘.
~ Similarly, in △DEF, the angles are ∠D + ∠E + ∠F = 180^∘.
Since ∠B = ∠E, their measures are identical, confirming that ∠E = 45^∘.

Final Answer: B) 45^∘​

Question: The function 𝑔(𝑥) = 20 + 5𝑥 represents the total cost, in dollars, of attending a carnival when 𝑥 rides are taken. How many rides can be taken for a total cost of $70?
A) 8 rides
B) 9 rides
C) 10 rides
D) 11 rides

Choice C is correct.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Step 1: Understand the given function
The function g(x) = 20 + 5x gives the total cost, where:
~ 20 is the fixed entry cost to attend the carnival.
~ 5x represents the cost of x rides, with each ride costing $5.
We are tasked with finding the number of rides (x) that can be taken for a total cost of $70.

Step 2: Set up the equation
Substitute the total cost into the equation g(x) = 70:
70 = 20 + 5x

Step 3: Solve for x
1. Subtract 20 from both sides to isolate the 5x-term:
70 − 20 = 5x
50 = 5x
2. Divide both sides by 5 to solve for x:
x = 50/5
x = 10

Step 4: Verify the solution
If x = 10, the total cost becomes:
g(x) = 20 + 5(10)
g(x) =20 + 50
g(x) = 70
This matches the total cost given in the problem.

Final Answer: C) 10 rides​

Question: 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?
A) 24 days
B) 26 days
C) 28 days
D) 32 days

Choice B is correct.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the absolute value of y, not the value of x.

Step 1: Understand the given information
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.

Step 2: Set up the equation
The number of books left after x days can be expressed as:
Remaining books = Initial inventory − (Books sold per day × Number of days)
Substituting the given information:
1,900 = 3,200 − 50x

Step 3: Solve for x
1. Subtract 1,900 from both sides to isolate the 3,200 − 50x−term:
1,900 − 3,200 = −50x
-1,300 = -50x
2. Divide both sides by −50 to solve for x:
x = −1,300/−50
x = 26

Step 4: Verify the solution
If x = 26, the remaining books are:
Remaining books = 3,200 − (50 × 26)
Remaining books =3,200 − 1,300
Remaining books = 1,900
This matches the target inventory.

Final Answer: B) 26 days

y > 4x + 8
Question: For which of the following tables are all the values of x and their corresponding values of y solutions to the given inequality?

Choice A is correct. In each choice, the values of x are 2, 4, and 6. Substituting the first value of x, 2, for x in the given inequality yields y > 4(2) + 8, or y > 16. Therefore, when x = 2, the corresponding value of y must be greater than 16. Of the given choices, only choice A is a table where the value of y corresponding to x = 2 is greater than 16. To confirm that the other values of x in this table and
their corresponding values of y are also solutions to the given inequality, the values of x and y in the table can be substituted for x and y in the given inequality. Substituting 4 for x and 30 for y in the given inequality yields 30 > 4(4) + 8, or 30 > 24, which is true. Substituting 6 for x and 41 for y in the
given inequality yields 41 > 4(6) + 8, or 41 > 32, which is true. It follows that for choice A, all the values of x and their corresponding values of y are solutions to the given inequality.

Choice B is incorrect. Substituting 2 for x and 8 for y in the given inequality yields 8 > 4(2) + 8, or 8 > 16, which is false.

Choice C is incorrect. Substituting 2 for x and 13 for y in the given inequality yields 13 > 4(2) + 8, or 13 > 16, which is false.

Choice D is incorrect. Substituting 2 for x and 13 for y in the given inequality yields 13 > 4(2) + 8, or 13 > 16, which is false.

Given Problem:
We need to verify for which of the given tables (A, B, C, or D) all the x and corresponding y values satisfy the inequality:
y > 4x + 8

Step-by-Step Solution:
Step 1: Understand the inequality
The inequality y > 4x + 8 requires that, for each pair of x and y, the value of y must be strictly greater than the calculated value of 4x + 8.

Step 2: Compute 4x + 8 for each x value in the tables
We will substitute the x values from each table into 4x + 8, then compare the resulting values with the corresponding y values to check whether the inequality is satisfied.

Option A:

We have already got our correct option but still, let’s try solving other options for better clearance.

Option B:

Option C:

Option D:

Final Answer:
The correct option is A, where all the x and y values satisfy the inequality y > 4x + 8.

Question: Which expression is equivalent to (x² + 11)² + (x − 5)(x + 5)?
A) x⁴ + 23x² − 14
B) x⁴ + 23x² + 96
C) x⁴ + 12x² + 121
D) x⁴ + x² + 146

Choice B is correct. The expression (x² + 11)² can be written as (x² + 11)(x² + 11), which is equivalent to x²(x² + 11) + 11(x² + 11). Distributing x² and 11 to (x² + 11) yields x⁴ + 11x² + 11x² + 121. The
expression (x – 5)(x + 5) is equivalent to (x – 5)x + (x – 5)5. Distributing x and 5 to (x – 5) yields x² – 5x + 5x – 25, or x² – 25. Therefore, the expression (x² + 11)² + (x – 5)(x + 5) is equivalent to (x⁴ + 22x² + 121) + (x² – 25), or x⁴ + 22x² + 121 + x² – 25. Combining like terms in this expression yields x⁴ + 23x² + 96.

Choice A is incorrect. Equivalent expressions must be equivalent for any value of x. Substituting 0 for x in this expression yields -14, whereas substituting 0 for x in the given expression yields 96.

Choice C is incorrect. Equivalent expressions must be equivalent for any value of x. Substituting 0 for x in this expression yields 121, whereas substituting 0 for x in the given expression yields 96.

Choice D is incorrect. Equivalent expressions must be equivalent for any value of x. Substituting 0 for x in this expression yields 146, whereas substituting 0 for x in the given expression yields 96.

Step-by-Step Solution
Step 1: Expand (x² + 11)²
Using the formula for the square of a binomial, (a + b)² = a² + 2ab + b², expand (x² + 11)²:
(x² + 11)² = (x²)² + 2(x²)(11) + 11².
Simplify each term:
~ (x²)² = x⁴,
~ 2(x²)(11) = 22x²
~ 11² = 121
Thus: (x² + 11)² = x⁴ + 22x² + 121.

Step 2: Expand (x − 5)(x + 5)
Using the difference of squares formula, (a − b)(a + b) = a² − b², expand (x − 5)(x + 5):
(x − 5)(x + 5) = x² − 5².
Simplify: x² − 5² = x² − 25.

Step 3: Add the expanded terms
Now add the results from Steps 1 and 2:
(x² + 11)² + (x − 5)(x + 5)
(x⁴ + 22x² + 121) + (x² − 25).
x⁴ + 22x² + 121 + x² − 25
x⁴ + 22x² + x² + 121 − 25
x⁴ + 23x² + 96
Group-like terms:
~ x⁴,
~ 22x² + x² = 23x²,
~ 121 − 25 = 96 (Positive 96 because 121 is a bigger value).
Thus: (x² + 11)² + (x − 5)(x + 5) = x⁴ + 23x² + 96.

Final Answer:
The equivalent expression is: B) x⁴ + 23x² + 96.

Question: What is the equation of the line that passes through the point (0, 5) and is parallel to the graph of
y = 7x + 4 in the xy-plane?
A) y = 5x
B) y = 7x + 5
C) y = 7x
D) y = 5x + 7

Choice B is correct. The equation of a line in the xy-plane can be written in slope-intercept form y = mx + b, where m is the slope of the line and (0, b) is its y-intercept. It’s given that the line passes through the point (0, 5). Therefore, b = 5. It’s also given that the line is parallel to the graph of y = 7x + 4, which means the line has the same slope as the graph of y = 7x + 4. The slope of the graph of y = 7x + 4 is 7. Therefore, m = 7. Substituting 7 for m and 5 for b in the equation y = mx + b yields y = 7x + 5.

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.

Choice C is incorrect. The graph of this equation passes through the point (0, 0), not (0, 5).

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.

We are tasked with finding the equation of a line that passes through the point (0, 5) and is parallel to the graph of y = 7x + 4.

Step-by-Step Solution
Step 1: Understand the slope of parallel lines
The slope of a line parallel to another line must be the same as the slope of the original line. The given line y = 7x + 4 is in slope-intercept form (y = mx + b), where m represents the slope.
From y = 7x + 4, the slope m = 7. Therefore, the equation of the new line will also have a slope of m = 7.

Step 2: Write the general form of the new line
The equation of a line with slope m = 7 can be written in slope-intercept form:
y = 7x + b,
where b is the y-intercept.

Step 3: Use the given point to find b
We know the line passes through the point (0, 5). Substituting x = 0 and y = 5 into the equation y = 7x + b, we solve for b:
5 = 7(0) + b.
Simplify: b = 5.
Thus, the equation of the line is:
y = 7x + 5.

Step 4: Verify parallelism
To confirm, the slope of the new line is m = 7, which is the same as the slope of the given line. Therefore, the two lines are indeed parallel.

Final Answer: B) y = 7x + 5.​

Question: In the linear function h, h(0) = 41 and h(1) = 40. Which equation defines h?
A) h(x) = −x + 41
B) h(x) = −x
C) h(x) = −41x
D) h(x) = −41

Choice A is correct. An equation defining a linear function can be written in the form h(x) = ax + b, where a and b are constants. It’s given that h(0) = 41. Substituting 0 for x and 41 for h(x) in the equation h(x) = ax + b yields 41 = a(0) + b, or b = 41. Substituting 41 for b in the equation h(x) = ax + b yields h(x) = ax + 41. It’s also given that h(1) = 40. Substituting 1 for x and 40 for h(x) in the equation h(x) = ax + 41 yields 40 = a(1) + 40, or 40 = a + 41. Subtracting 41 from the left and right-hand sides of this equation yields -1 = a. Substituting -1 for a in the equation h(x) = ax + 41 yields h(x) = -1x + 41, or h(x) = -x + 41.

Choice B is incorrect. Substituting 0 for x and 41 for h(x) in this equation yields 41 = -0, which isn’t a true statement.

Choice C is incorrect. Substituting 0 for x and 41 for h(x) in this equation yields 41 = -41(0), or 41 = 0, which isn’t a true statement.

Choice D is incorrect. Substituting 41 for h(x) in this equation yields 41 = -41, which isn’t a true statement.

Step 1: Understand the structure of a linear equation
A linear function is typically written in the form:

Step 3: Write the equation
Now substitute m = −1 and b = 41 into the equation:
h(x) = −x + 41.

Step 4: Verify the solution
Substitute x = 0 and x = 1 into the equation to confirm it matches the given values:
~ For x = 0:
h(0) = −0 + 41 = 41(correct).
~ For x = 1:
h(1) = −1 + 41 = 40(correct).
Thus, the equation h(x) = −x + 41 is correct.

Final Answer: A) h(x) = −x + 41

Question: Which expression is equivalent to 6x⁸y² + 12x²y²?
A) 6x²y²(2x⁶)
B) 6x²y²(x⁴)
C) 6x²y²(x⁶ + 2)
D) 6x²y²(x⁴ + 2)

Choice C is correct. Since each term of the given expression has a common factor of 6x²y², it may be rewritten as 6x²y²(x⁶) + 6x²y²(2), or 6x²y²(x⁶ + 2).

Choice A is incorrect. This expression is equivalent to 12x⁸y², not 6x⁸y² + 12x²y².

Choice B is incorrect. This expression is equivalent to 6x⁶y², not 6x⁸y² + 12x²y².

Choice D is incorrect. This expression is equivalent to 6x⁶y² + 12x²y², not 6x⁸y² +12x²y².

Step 1: Identify the common factors
Both terms have the following factors in common:
~ The coefficient 6 (as both 6 and 12 are divisible by 6).
~ The variable x² (as the smallest power of x is x²).
~ The variable y² (as both terms include y²).
Thus, the greatest common factor (GCF) is 6x²y².

Step 2: Factor out the GCF

Final Answer: C) 6x²y²(x⁶ + 2)

Question: 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 2x + 35y = 3,934 represents this situation. Which of the following is the best interpretation of x in this context?
A) The average number of trees per hectare in the park
B) The average number of trees per hectare in the residential area
C) The total number of trees in the park
D) The total number of trees in the residential area

Choice A is correct. It’s 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’s also given that the equation 2x + 35y = 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 2x and 35y correspond to the number of trees in the park and in the residential area, respectively. Since 2x corresponds to the number of trees in the park, and 2 is the size of the park, in hectares, x must represent the average number of trees per hectare in the park.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.

Choice D is incorrect and may result from conceptual errors.

Solution
We are tasked with interpreting the variable x in the equation 2x + 35y = 3,934, which represents the total number of trees in a neighborhood. Let’s break this down step-by-step.

Step 1: Understand the given equation
The equation is: 2x + 35y = 3,934
This represents:
~ 2x: The number of trees in the 2-hectare park.
~ 35y: The number of trees in the 35-hectare residential area.
~ 3,934: The total number of trees in the neighborhood.
Thus, x and y are the average numbers of trees per hectare in the park and residential area, respectively.

Step 2: Analyze x
From the term 2x, we see that x is multiplied by 2 (the size of the park in hectares). Therefore, x must represent the number of trees per hectare in the park.
Multiplying x by 2 gives the total number of trees in the 2-hectare park. Hence, x is not the total number of trees in the park, nor is it related to the residential area.

Step 3: Eliminate incorrect options
Let’s analyze each option:
~ Option A: The average number of trees per hectare in the park
This is correct, as x represents the number of trees per hectare in the 2-hectare park.
~ Option B: The average number of trees per hectare in the residential area
This is incorrect, as y, not x, represents the number of trees per hectare in the residential area.
~ Option C: The total number of trees in the park
This is incorrect, as 2x, not x, represents the total number of trees in the park.
~ Option D: The total number of trees in the residential area
This is incorrect, as 35y, not x, represents the total number of trees in the residential area.

Step 4: Verify the interpretation
To confirm: x is the average number of trees per hectare in the park, which, when multiplied by 2 (the size of the park), gives the total number of trees in the park.

Final Answer: A) The average number of trees per hectare in the park

Question: The graph shows the relationship between the number of shares of stock from Company A, x, and the number of shares of stock from Company B, y, that Simone can purchase. Which equation could represent this relationship?
A) y = 8x + 12
B) 8x + 12y = 480
C) y = 12x + 8
D) 12x + 8y = 480

Choice B is correct. The graph shown is a line passing through the points (0, 40) and (60, 0). Since the relationship between x and y is linear, if two points on the graph make a linear equation true, then the equation represents the relationship. Substituting 0 for x and 40 for y in the equation in choice B,
8x +12y = 480, yields 8(0) + 12(40) = 480, or 480 = 480, which is true. Substituting 60 for x and 0 for y in the equation 8x +12y = 480 yields 8(60) + 12(0) = 480, or 480 = 480, which is true. Therefore, the equation 8x +12y = 480 represents the relationship between x and y.

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.

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.

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.

Step-by-Step Solution:
Step 1: Understand the relationship
The graph represents the relationship between the number of shares Simone can purchase from Company A (x) and Company B (y). This relationship is linear, meaning it can be expressed as an equation of the form:
ax + by = c
where a, b, and c are constants determined by the context.

Step 2: Interpret the graph’s key points
From the graph:
~ When x = 0 (Simone buys no shares of Company A), y = 40 (she buys 40 shares of Company B).
~ When y = 0 (Simone buys no shares of Company B), x = 60 (she buys 60 shares of Company A).
These points are (0, 40) and (60, 0), which are intercepts of the line.

Step 4: Verify the equation
To confirm, substitute the intercepts: ax + by = c
where a = 8, b = 12, c = 480, and (x, y) = (0, 40) or (60, 0).
1. For x = 0:
8(0) + 12y = 480
 ⟹  12y = 480
 ⟹  y = 480/12
 ⟹  y = 40
This matches the graph (x, y) = (0, 40). If x = 0 then y = 40.
2. For y = 0:
8x + 12(0) = 480
⟹  8x = 480
⟹  x = 480/8
⟹  x = 60
This also matches the graph (x, y) = (60, 0). If y = 0 then x = 60.
Thus, the equation 8x + 12y = 480 correctly represents the relationship.

Question: 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’s cost for the bag. What was the store’s cost, in dollars, for the bag?
A) $20
B) $24
C) $25
D) $30

Choice C is the correct answer.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Expanded Step-by-Step Solution
Step 1: Understanding the problem
The question describes a relationship between the regular price, the sale price, and the store’s cost:
1. The regular price of the bag is $60.
2. The sale price is 50% less than the regular price.
3. The sale price is 20% greater than the store’s cost.
The goal is to calculate the store’s cost based on these relationships.

Step 2: Calculating the sale price
The sale price is 50% less than the regular price. To find this:
1. Start by calculating 50% of the regular price. Recall that 50% means half of the value or multiplying the value by 0.5.
50% × 60
0.5 × 60 = 30
2. Subtract this value from the regular price:
Sale Price = Regular Price − (50% × Regular Price)
Sale Price = 60 − 30 = 30
Thus, the sale price is $30.

Step 3: Relating the sale price to the store’s cost
The problem states that the sale price is 20% greater than the store’s cost. To express this mathematically:
1. Let the store’s cost be x.
2. The sale price is 20% more than x. Add 20% of x to x:
Sale Price = x + 0.2x
Combine like terms:
Sale Price = 1.2x
The equation becomes:
30 = 1.2x

Step 5: Verifying the solution
Let’s confirm the calculation:
1. If the store’s cost is $25, then 20% of $25 is:
20% × 25
0.2 × 25 = 5
2. Add this value to the store’s cost:
25 + 5 = 30
The sale price is $30, matching the earlier calculation. This confirms the solution.

Final Answer: C) $25

Question: Kaylani used fabric measuring 5 yards in length to make each suit for a men’s choir. The relationship between the number of suits that Kaylani made, x, and the total length of fabric that she purchased y, in yards, is represented by the equation y − 5x = 6. What is the best interpretation of 6 in this context?
A) Kaylani made 6 suits.
B) Kaylani purchased a total of 6 yards of fabric.
C) Kaylani used a total of 6 yards of fabric to make the suits.
D) Kaylani purchased 6 yards more fabric than she used to make the suits.

Choice D is correct. It’s given that the equation y – 5x = 6 represents the relationship between the number of suits that Kaylani made, x, and the total length of fabric she purchased, y, in yards. Adding 5x to both sides of the given equation yields y = 5x + 6. Since Kaylani made x suits and used 5 yards of fabric to make each suit, the expression 5x represents the total amount of fabric she used to make the suits. Since y represents the total length of fabric Kaylani purchased, in yards, it follows from the equation y = 5x + 6 that Kaylani purchased 5x 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.

Choice A is incorrect. Kaylani made a total of x suits, not 6 suits.

Choice B is incorrect. Kaylani purchased a total of y yards of fabric, not a total of 6 yards of fabric.

Choice C is incorrect. Kaylani used a total of 5x yards of fabric to make the suits, not a total of 6 yards of fabric.

Problem Analysis
The given equation is: y − 5x = 6
Here:
~ y represents the total length of fabric Kaylani purchased (in yards).
~ x represents the number of suits Kaylani made.
~ Each suit requires 5 yards of fabric.
We need to interpret the constant 6 in the equation.

Step-by-Step Solution
Step 1: Rearrange the equation
Rearrange the given equation to isolate y (the total fabric purchased): y = 5x + 6
From this form, we see:
~ 5x represents the total fabric required to make x suits, where each suit uses 5 yards.
~ 6 is an additional amount of fabric purchased beyond the amount needed for making the suits.

Step 2: Interpret 6
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 extra fabric purchased.

Step 3: Verify with an example
Let’s verify the interpretation by substituting values for x.
1. If Kaylani makes x = 0 suits, the equation becomes:
y = 5(0) + 6
y = 6
This means she purchased 6 yards of fabric even when no suits were made.
2. If Kaylani makes x = 2 suits, the equation becomes:
y = 5(2) + 6
y =10 + 6
y =16
This means she purchased 16 yards of fabric, 10 of which were used for the suits, and 6 yards were extra.
Thus, 6 is the extra fabric purchased.

Step 4: Identify the correct option
From the given options:
~ A) “Kaylani made 6 suits” is incorrect because 6 is not related to the number of suits made.
~ B) “Kaylani purchased a total of 6 yards of fabric” is incorrect because 6 is the extra fabric, not the total fabric.
~ C) “Kaylani used a total of 6 yards of fabric to make the suits” is incorrect because the fabric used for suits is 5x, not 6.
~ D) “Kaylani purchased 6 yards more fabric than she used to make the suits” is correct because 6 represents the extra fabric purchased.
The correct answer is: Option D​

Question: For x > 0, the function g is defined as follows: g(x) equals 125% of x. Which of the following could describe this function?
A) Decreasing exponential
B) Decreasing linear
C) Increasing exponential
D) Increasing linear

Choice D is the correct answer.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Step-by-Step Solution:
Step 1: Understanding the function definition
The function g(x) is defined as 125% of x.
This means: g(x) = 125% multiplying by x
To convert the percentage into a decimal form:
125% = 125/100
125% = 1.25
Thus, the function becomes: g(x) = 1.25x
This represents a linear relationship because g(x) is proportional to x, with the constant multiplier 1.25.

Step 2: Identifying the function type
1. Linear nature:
The equation g(x) = 1.25x follows the general form of a linear function:
y = mx + b
Here, m = 1.25 (the slope), and b = 0 (no y-intercept other than the origin).
2. Increasing behavior:
Since m = 1.25 > 0, the function increases as x increases.
3. Not exponential:
An exponential function has the form y = ab^x, where b≠1. The function g(x) = 1.25x does not fit this form.

Step 3: Eliminating incorrect options
Let’s analyze each option:
A) Decreasing exponential: Incorrect. The function is not exponential and does not decrease.
B) Decreasing linear: Incorrect. Although the function is linear, it is increasing because m > 0.
C) Increasing exponential: Incorrect. While the function increases, it is not exponential.
D) Increasing linear: Correct. The function increases linearly with a constant slope of 1.25.

Step 4: Verification
For x > 0, substitute sample values of x to confirm the behavior:
~ If x = 1, then g(x) = 1.25(1) = 1.25.
~ If x = 2, then g(x) = 1.25(2) = 2.5.
~ If x = 3, then g(x) = 1.25(3) = 3.75.
As x increases, g(x) increases proportionally, confirming the linear nature of the function.

Final Answer: D) Increasing linear

Question: A rectangular park has an area of 3.25 square miles. What is the area, in square yards, of this park? (1 mile = 1,760 yards)
A) 8,540
B) 620,480
C) 5,004,800
D) 9,984,400

Choice C is the correct answer.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Step-by-Step Solution
Step 1: Understanding the problem
We are tasked with converting an area from square miles to square yards. The given area is 3.25 square miles.
We know:
1 mile = 1,760 yards.
To find the area in square yards, we first convert the unit of miles to yards and then square the result.

Step 2: Converting miles to yards
Since 1 mile = 1,760 yards, the length of one side of a square mile is:
1 mile = 1,760 yards
To find the area in square yards, square this conversion factor:
(1 mile)² = (1,760 yards)²
(1 mile)² = 1,760 × 1,760 = 3,097,600 square yards.
Thus, 1 square mile = 3,097,600 square yards.

Step 3: Multiply by the given area in square miles
The park has an area of 3.25 square miles.
To find the area in square yards, multiply 3.25 by 3,097,600:
3.25 × 3,097,600 = 10,466,200 square yards.

Step 4: Verification of Calculation
Break down the multiplication for verification:
3.25 × 3,097,600 = (3 × 3,097,600) + (0.25 × 3,097,600).
~ Compute 3 × 3,097,600:
3 × 3,097,600 = 9,292,800.
~ Compute 0.25 × 3,097,6000:
0.25 × 3,097,600 = 774,400.
~ Add the two results:
9,292,800 + 774,400 = 10,466,200.
Thus, the total area is correctly calculated as 10,466,200 square yards.

Question: In the xy-plane, a parabola has vertex (6, −10) and intersects the x-axis at two points. If the equation of the parabola is written in the form y = ax² + bx + c, where a, b, and c are constants, which of the following could be the value of a + b + c?
A) −15
B) 15
C) 37
D) -37

Choice B is correct.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Step-by-Step Solution
Step 1: Understand the Vertex Form of the Parabola
The equation of a parabola in vertex form is:
y = a(x − h)² + k,
where (h, k) is the vertex.
For this parabola, the vertex is given as (6, −10), so:
y = a(x − 6)² − 10.

Step 2: Determine the Standard Form
Expand y = a(x − 6)² − 10 to convert it to standard form:
It means (x – 6)² gives algebric formula (a – b)² = (a² – 2ab + b²). Hence: y = a(x² − 12x + 36) − 10.
Distribute a:
y = ax² − 12ax + 36a − 10.
This gives the standard form:
y = ax² − 12ax + (36a − 10).
From the standard form, the coefficients are:
~ a = a,
~ b = −12a,
~ c = 36a − 10.

Step 3: Use the Condition for Intersection with the x-Axis
For the parabola to intersect the x-axis at two points, the discriminant of the quadratic equation must be positive (Δ > 0):
Δ = b² − 4ac.
Substitute b = −12a and c = 36a − 10 into the discriminant:
Δ = (−12a)² −4(a)(36a − 10).
Simplify: Δ = 144a² − 4(36a² − 10a).
Δ = 144a² − 144a² + 40a.
Δ = 40a.
Since Δ > 0, a > 0.

Step 4: Solve for a + b + c
Choose a simple value for a (e.g., a = 1, as a > 0):
~ b = −12a
b = −12(1)
b = −12
~ c = 36a − 10
c = 36(1) − 10
c = 36 – 10
c = 26.
Now calculate a + b + c:
a + b + c
1 − 12 + 26 =15.
Thus, a + b + c = 15.

Verification:
Substitute back into the equation and check if the parabola still satisfies the conditions (e.g., correct vertex and intersects the x-axis):
~ Equation: y = (x − 6)2 − 10, expanded as y = x2 − 12x + 26.
~ Vertex is (6, −10), as required, and discriminant Δ > 0, confirming intersection with the x-axis.

Final Answer: B) 15

Question: The number p is 85% greater than the number q. The number q is 75% less than 60. What is the value of p?
A) 15
B) 15.75
C) 25.25
D) 27.75

Choice D is the correct answer.

Choice A is incorrect and may result from conceptual or calculation errors.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Step-by-Step Solution
Step 1: Calculate the Value of q
The problem states that q is 75% less than 60. To find q:
q = 60 − (75%  of 60)
q = 60 − 0.75 × 60
q = 60 − 45
Thus, q = 15.

Step 2: Calculate the Value of p
The problem states that p is 85% greater than q. This means p is equal to q plus 85% of q. To find p:
p = q + (85% of q)
p = q + 0.85 × q
p = 15 + 0.85 × 15
p = 15 + 12.75 = 27.75
Thus, p = 27.75.

Verification
~ Calculated q = 15 as 75% less than 60.
~ Calculated p = 27.75 as 85% greater than q.
Both steps are consistent with the problem’s conditions.

Final Answer: D) 27.75

Question: The function g is defined by g(x) = x(x − 2)(x + 6)². The value of g(7− w)is 0, where w is a constant. What is the sum of all possible values of w?
A) 25
B) 7
C) 5
D) 13
[Type-Based Answer: In the final exam, you will type the answer rather than choose from options.]

Option A: The correct answer is 25. The value of g(7 – w) is the value of g(x) when x = 7 – w, where w is a constant. Substituting 7 – w for x in the given equation yields g(7 – w) = (7 – w)(7 – w – 2)(7 – w + 6)², which is equivalent to g(7 – w) = (7 – w)(5 – w)(13 – w)². It’s given that the value of g(7 – w) is 0. Substituting 0 for g(7 – w) in the equation g(7 – w) = (7 – w)(5 – w)(13 – w)² yields 0 = (7 – w)(5 – w)(13 – w)². 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 w can be found by setting each factor equal to 0. Setting the first factor equal to 0 yields 7 – w = 0. Adding w to both sides of this equation yields 7 = w. Therefore, 7 is one possible value of w. Setting the second factor equal to 0 yields 5 – w = 0. Adding w to both sides of this equation yields 5 = w. Therefore, 5 is a second possible value of w. Setting the third factor equal to 0 yields (13 – w)² = 0. Taking the square root of both sides of this equation yields 13 – w = 0. Adding w to both sides of this equation yields 13 = w. Therefore, 13 is a third possible value of w. Adding the three possible values of w yields 7 + 5 + 13, or 25. Therefore, the sum of all possible values of w is 25.

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect and may result from conceptual or calculation errors.

Problem Analysis:
The given function is:
g(x) = x(x − 2)(x + 6)²
The question states that g(7 − w) = 0, and we are to find the sum of all possible values of w.
For g(7 − w) to equal 0, any of the factors of g(x) must be 0 because a product is 0 if at least one of its factors is 0.

Step-by-Step Solution:
Step 1: Replace x with 7 − w
Substitute 7 − w for x in g(x): x(x − 2)(x + 6)²
g(7 − w) = (7 − w)[(7 − w) − 2][(7 − w) + 6]²
Simplify each term:
~ The first factor: 7 − w
~ The second factor: (7 − w) − 2
7 – 2 – w
5 − w
~ The third factor: (7 − w) + 6
7 + 6 – w
13 − w
So:
g(7 − w) = (7 − w)(5 − w)(13 − w)²

Step 2: Solve for g(7 − w) = 0
For g(7 − w) = 0, at least one of the factors must equal 0. Solve each factor:
~ First factor: 7 − w = 0 ⟹  w = 7
~ Second factor: 5 − w = 0  ⟹  w = 5
~ Third factor: 13 − w = 0  ⟹  w = 13

Step 3: Find the sum of all possible values of w
~ The possible values of w are:
w = 7, w = 5, w = 13
~ The sum of these values is: 7 + 5 + 13 = 25

Final Answer:
The sum of all possible values of w is: A) 25​