QUESTION EXPLANATIONS

1. (B) — We can multiply 3x = 15 by 2 to get 6x = 30. Similarly, multiplying 2y = 10 by 2 gives us 4y = 20. 6x + 4y = 30 + 20 = 50.

2. (D) — f(x) = 2x² – 8x + 6 describes an upwards-opening parabola with roots at x = 1 and x = 3, which we can find by setting the equation equal to 0 and factoring. Therefore, any x-value between but not including 1 and 3 will be less than zero; 2 is the only option that satisfies this condition. Plugging this value into the equation confirms that at this point f(x) is less than zero.

3. (D) — We can find the slope of line p by using the formula slope = rise/run. Rise equals the difference between the y values (9 – 3 = 6), and run is the difference between the x values (5 – 2 = 3). Therefore, slope = 6/3 = 2. Answer D) is the only solution with the correct slope.

4. (B) — If the printer can produce 40 tank tops in 5 hours, they can produce 8 tank tops in one hour. Therefore they can make 8 × 7 = 56 tank tops in 7 hours. You can also set up the proportion 40/5 = x/7 and solve for x.

5. (A) — The graph shows the basic absolute value graph shifted in the negative y direction. To shift the function down (that is, in the negative y direction), you must subtract a quantity outside of the absolute value signs. Answer A) is the only equation that does this.

6. (C) — Since there are 180 degrees in a triangle, we can find the third angle in the top triangle by calculating 180 – (75 + 75) = 30 degrees. Because opposite angles are always congruent, we know that the top angle in the larger triangle is also 30 degrees. That means that the larger triangle is a 30-60-90 triangle, so its hypotenuse (x) is twice the length of its shorter side — in this case, 5. Twice 5 is 10.

7. (B) — Answer A) describes linear growth; answer B) describes exponential growth. Exponential growth is always faster than linear growth, so A) is incorrect. Answer C) describes a power function (because the exponent is constant); exponential growth, where the exponent is a variable, also increases faster than a power function. Finally, answer D) is another linear equation. Answer B) grows the fastest.

8. (C) — f(-1) = 3(-1) + 7 = 4
f(3) = 3(3) + 7 = 16
4 + 16 = 20

9. (C) — Using the FOIL method, we get 2i² + 11i + 12. Since i = √ -1 , i² = -1, so the expression is equal to -2 + 11i + 12 = 10 + 11i.

10. (B) — Although this problem can quickly be solved using guess-and-check, we can also multiply both sides by 3 to get 6x + x = 15x or 7x = 15x. It’s clear at this point that the only possible value of x that will satisfy this equation is 0.

11. (C) — The fastest way to find the answer is to plug numbers into the given equations. Answer A) can be eliminated by plugging in 1 for x. Answer B) can be eliminated by plugging in 2 for x. Answer D) can be eliminated by plugging in 0 for x. This leaves answer C), which satisfies all values in the table.

12. (C) — Squaring both sides gives 5x – 5 = y² + 1 and rearranging gives y² = 5x – 6. Plugging in each pair of values shows that only answer C) is correct (for example, plugging in (1, 2) for x and y gives 22 = 5(1) – 6, or 4 = 1, which is false, but plugging in (3, 3) gives 32 = 5(3) – 6, or 9 = 9).

13. (A) — g(5) = f(15) = 152 + 2(15) + 3 = 225 + 30 + 3 = 258
f(5) = 52 + 2(5) + 3 = 25 + 10 + 3 = 38
258 – 38 = 220

14. (A) — We can rearrange the second equation to get x = 35 – 2y. Plugging this into the first equation gives us 2(35 – 2y) + y = 10, or 70 – 4y + y = 10. Simplifying gives 3y = 60, or y = 20. Alternatively, you can multiply either equation by -2 and add them. It’s fastest to eliminate x by doing this to the bottom equation: -2x – 4y = -70. Adding this to the top equation gives -3y = -60, so y = 20.

15. (B) — The pattern involves adding the next integer in the sequence 1, 2, 3, 4… to the previous term in the given sequence. We start with 1 + 1 = 2. We take this answer and add 2: 2 + 2 = 4. We take this answer in turn and add 3: 4 + 3 = 7, then take that answer and add 4: 7 + 4 = 11. Continuing this pattern gives us 11 + 5 = 16.

16. (10) — The ratio of chickens to land is 5:1.5, so multiplying by 2 gives us 10:3 – Jeff and Liz will purchase 10 chickens for 3 acres of land.

17. (5) — Rewriting the bottom equation gives us x = 4 + y. Plugging this into the top equation gives (4 + y)² – y² = 20, or 16 + 8y + y² – y² = 20. Simplifying and rearranging gives 8y = 4, so y = ½. Plugging this into either equation gives us x = 4½, so x + y = ½ + 4½ = 5. Alternatively, we can rewrite the first equation as (x – y)(x + y) = 20. Since x – y = 4, x + y must equal 5.

18. (0) — Plugging the first equation into the second gives us y = -2(2y + 6) – 3 = -4y – 12 – 3. Simplifying and rearranging gives 5y = -15, so y = -3. Plugging this into the first equation gives x = 2(-3) + 6 = 0, so xy = (0)(-3) = 0.

19. (16) — If we double a variable that is then squared, the effect is to multiply that squared variable by 4: (2x)² = 4x². The result of this operation is to reduce the force to one-fourth of its previous value, since the change is to the denominator. 64/4 = 16. We can also plug the given values into the equations to give us 64 = x/42, where x is G Χ m₁ Χ m₂, which do not change. Solving for x gives us x = 1024. To find the force of gravity, we can write F = 1024/8² = 1024/64 = 16.

20. (48) — The perimeter of a triangle is the sum of the lengths of its sides. The base is the x-value when y = 0. y = 0 = -4x/3 + 16, so x = 12. The left side is the y-value when x = 0. y = -4(0)/3 + 16 = 16. We can draw this as below:

The triangle is a 3-4-5 triangle, with each side multiplied by 4, so the hypotenuse is 5 Χ 4 = 20 (the hypotenuse can also be found from the Pythagorean Theorem: c² = a² + b² = 122 + 162 = 144 + 256 = 400, so c = 20). 12 + 16 + 20 = 48.