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QUESTION EXPLANATIONS

For Test (Math Test - No Calculator)

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1. (C) — The expression given to you in the question is the cost of a monthly texting plan in dollars. No matter how many texts, t, are sent, the cost is still 20 dollars a month in addition to a cost that is dependent on the number of text messages sent.

(A) is incorrect; each text message costs 10 cents.
(D) is also incorrect; the amount the customer pays increases for each text sent.

 
2. (C) — You can see that the slope of the function is negative, since the height of the function decreases as x increases. Therefore, you can eliminate choices (A) and (B). The function intersects the y-axis at a value above the x-axis, so you know the y-intercept must be greater than 0. This allows you to eliminate choice (D), leaving you with the correct answer, (C).
 
3. (A) — To solve for a – b, you first need to find the value of both variables! Do this by expanding the left-hand side of the equation given to you:

You can see that a = 15, the coefficient of x2, and b = 13, the coefficient of x. Therefore, a – b = 2.
 
4. (B) — You know that the ruler length cannot be larger or smaller than 1 meter by more than 1 millimeter, or 0.001 of a meter. The expression |x – 1| represents how much the length of the ruler differs from 1 meter, so |x – 1| ≤ 0.001.

If you chose (A), you may have converted to centimeters instead of meters.

 
5. (B) — The question is asking you to find an equation that relates the number of nitric oxide molecules and the number of water molecules produced by the reaction. The reaction always produces 2 nitric acid molecules for every 3 water molecules, so the ratio of nitric acid to water molecules is $2:3$. At this point, you may have noticed the similarity of this ratio to the equation $2n = 3w$, answer choice (A). However, this equation is actually backwards! The ratio shows us that 3 times the number of nitric acid molecules is equal to 2 times the number of water molecules, or $3n = 2w$. For example, if the reaction produced 4 molecules of nitric acid, it would need to produce 6 molecules of water in order to satisfy the $2:3$ ratio. Plugging these in to the equation $3n = 2w$, you get $3(4) = 2(6)$, which is correct.
 
6. (B) — To find the two solutions, expand the left-hand side and solve for n. Remember that the right-hand side is not zero, so you still have some work to do!

You have found that n = –9 or n = 1, so the product of these two solutions is –9 Χ 1 = –9.

If you got (C), you forgot that the right-hand side was 36 and not zero, so you thought the solutions were n = 0 and n = –8.

 
7. (C) — Square both sides and solve for x:

x + 1 = 36
x = 35

If you got (A), you ignored the square root.
 
8. (A) — The number lines below show the range of possible values for x given the inequality in the question. Therefore, your first step should be isolating x:

2x ≥ 8
x ≥ 4

Now that you know x must be greater than or equal to 4, you can see that the only number line demonstrating this is choice (A).
 
9. (C) — First, notice the equation of line f (x) is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. The line g (x) is perpendicular to f (x) = 2x – 1, so its slope is the negative reciprocal of 2, or –1⁄2 . You also know that g (x) crosses the point (0, 4), so it has a y-intercept of 4. Now, set the two lines equal to each other and solve for the x-coordinate of their intersection:

The only answer choice with an x-value of 2 is (C), so you don’t need to solve for y.
 
10. (A) — You are given the equation of the function f (x), so to solve for a, plug (a + 4) into the function and solve:


 
11. (C) — You know that the formula for area of a triangle is Area = 1⁄2 base Χ height. The question gives you both the area and base in terms of x, so plug these expressions into the formula for area, and solve for height:

 
12. (C) — You can rewrite the expression as follows. Remember that when you are multiplying expressions with the same base, you simply add the exponents.

You cannot simplify this any further, since there is no rule for adding terms with the same base, so this is the final answer.
 
13. (C) — You know that Graham’s slice of pie is one eighth of the total pie, because 45° = 1⁄8 Χ 360°. Therefore, the circumference of the whole pie is 8π. Since the formula for calculating circumference of a circle is 2πr, you can see that 2r = 8, so r = 4.
 
14. (A) — Pictured above the question is an upwards-opening parabola that has been vertically shifted up by some number of units. Answer choices (C) and (D) are parabolas that have been horizontally shifted either to the right or left, so these are incorrect. Option (B) describes an upwards-facing parabola that has been shifted down, so it is also wrong. You are left with the correct answer, (A), which is indeed an upwards-facing parabola shifted up.
 
15. (B) — First, factor the numerator of the expression, then cancel out common factors of the numerator and the denominator:

 
16. (4 ≤ <i>y</i> ≤ 6) — The first step in solving an absolute value inequality is to remove the absolute value signs. By doing this you get two cases: either y – 5 ≤ 1, or y – 5 ≥ –1. Solve both by isolating for y and input any possible solution as your grid-in:

 
17. (2) — Simply isolate for x in the given equation:

15x – 6 = 3x + 18
12x = 24
x = 2


 
18. (4) — Let the number of students be represented by s. If there were 10 participants in total, then the number of non-students is equal to 10 – s. The study paid $120 in total, so you can write that 15s + 10(10 – s) = 120. Isolate for s to find the number of students:

 
19. (21) — Multiply both sides of the equation by the denominator to get rid of the fraction, then factor and solve for n:

Since n is equal to either 21 or 5, the greatest value that satisfies the equation is 21.
 
20. (1) — First, write out the equation given to you in the question, that is “a + bi can be multiplied by i to get 1:”

To go from the second line to the third line above, you need to use the fact that i2 = –1. For this equation to be true, you know that ai – b must equal 1. You can equate the coefficients: -b + ai = 1 + 0i, meaning that b = -1 and a = 0. Finally, a - b = 0 - (-1) = 1.



The expression given to you in the question is the cost of a monthly texting plan in dollars. No matter how many texts, t, are sent, the cost is still 20 dollars a month in addition to a cost that is dependent on the number of text messages sent.

(A) is incorrect; each text message costs 10 cents.
(D) is also incorrect; the amount the customer pays increases for each text sent.

You can see that the slope of the function is negative, since the height of the function decreases as x increases. Therefore, you can eliminate choices (A) and (B). The function intersects the y-axis at a value above the x-axis, so you know the y-intercept must be greater than 0. This allows you to eliminate choice (D), leaving you with the correct answer, (C).
To solve for a – b, you first need to find the value of both variables! Do this by expanding the left-hand side of the equation given to you:

You can see that a = 15, the coefficient of x2, and b = 13, the coefficient of x. Therefore, a – b = 2.
You know that the ruler length cannot be larger or smaller than 1 meter by more than 1 millimeter, or 0.001 of a meter. The expression |x – 1| represents how much the length of the ruler differs from 1 meter, so |x – 1| ≤ 0.001.

If you chose (A), you may have converted to centimeters instead of meters.

The question is asking you to find an equation that relates the number of nitric oxide molecules and the number of water molecules produced by the reaction. The reaction always produces 2 nitric acid molecules for every 3 water molecules, so the ratio of nitric acid to water molecules is $2:3$. At this point, you may have noticed the similarity of this ratio to the equation $2n = 3w$, answer choice (A). However, this equation is actually backwards! The ratio shows us that 3 times the number of nitric acid molecules is equal to 2 times the number of water molecules, or $3n = 2w$. For example, if the reaction produced 4 molecules of nitric acid, it would need to produce 6 molecules of water in order to satisfy the $2:3$ ratio. Plugging these in to the equation $3n = 2w$, you get $3(4) = 2(6)$, which is correct.
To find the two solutions, expand the left-hand side and solve for n. Remember that the right-hand side is not zero, so you still have some work to do!

You have found that n = –9 or n = 1, so the product of these two solutions is –9 Χ 1 = –9.

If you got (C), you forgot that the right-hand side was 36 and not zero, so you thought the solutions were n = 0 and n = –8.

Square both sides and solve for x:

x + 1 = 36
x = 35

If you got (A), you ignored the square root.
The number lines below show the range of possible values for x given the inequality in the question. Therefore, your first step should be isolating x:

2x ≥ 8
x ≥ 4

Now that you know x must be greater than or equal to 4, you can see that the only number line demonstrating this is choice (A).
First, notice the equation of line f (x) is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept. The line g (x) is perpendicular to f (x) = 2x – 1, so its slope is the negative reciprocal of 2, or –1⁄2 . You also know that g (x) crosses the point (0, 4), so it has a y-intercept of 4. Now, set the two lines equal to each other and solve for the x-coordinate of their intersection:

The only answer choice with an x-value of 2 is (C), so you don’t need to solve for y.
You are given the equation of the function f (x), so to solve for a, plug (a + 4) into the function and solve:


You know that the formula for area of a triangle is Area = 1⁄2 base Χ height. The question gives you both the area and base in terms of x, so plug these expressions into the formula for area, and solve for height:

You can rewrite the expression as follows. Remember that when you are multiplying expressions with the same base, you simply add the exponents.

You cannot simplify this any further, since there is no rule for adding terms with the same base, so this is the final answer.
You know that Graham’s slice of pie is one eighth of the total pie, because 45° = 1⁄8 Χ 360°. Therefore, the circumference of the whole pie is 8π. Since the formula for calculating circumference of a circle is 2πr, you can see that 2r = 8, so r = 4.
Pictured above the question is an upwards-opening parabola that has been vertically shifted up by some number of units. Answer choices (C) and (D) are parabolas that have been horizontally shifted either to the right or left, so these are incorrect. Option (B) describes an upwards-facing parabola that has been shifted down, so it is also wrong. You are left with the correct answer, (A), which is indeed an upwards-facing parabola shifted up.
First, factor the numerator of the expression, then cancel out common factors of the numerator and the denominator:

The first step in solving an absolute value inequality is to remove the absolute value signs. By doing this you get two cases: either y – 5 ≤ 1, or y – 5 ≥ –1. Solve both by isolating for y and input any possible solution as your grid-in:

Simply isolate for x in the given equation:

15x – 6 = 3x + 18
12x = 24
x = 2


Let the number of students be represented by s. If there were 10 participants in total, then the number of non-students is equal to 10 – s. The study paid $120 in total, so you can write that 15s + 10(10 – s) = 120. Isolate for s to find the number of students:

Multiply both sides of the equation by the denominator to get rid of the fraction, then factor and solve for n:

Since n is equal to either 21 or 5, the greatest value that satisfies the equation is 21.
First, write out the equation given to you in the question, that is “a + bi can be multiplied by i to get 1:”

To go from the second line to the third line above, you need to use the fact that i2 = –1. For this equation to be true, you know that ai – b must equal 1. You can equate the coefficients: -b + ai = 1 + 0i, meaning that b = -1 and a = 0. Finally, a - b = 0 - (-1) = 1.