Translate each statement, piece by piece. The first part tells us that “the product of x and y is 76.” Since product means multiplication, the first equation must be xy = 76, so you can eliminate (C). The second part says that “x is twice the square of y,” which translates to x = 2y², so eliminate (B) and (D), and (A) is the only choice left. Notice that only the y needs to be squared, which is why (B) is wrong. The second equation for (B) would be written as “the square of twice y,” which is not what the problem stated.

Notice that this question is asking for an expression instead of a variable, so manipulate the inequality to so that you get 4r + 3 in the inequality. Treat each side of the inequality separately to avoid confusion. Starting with the -6 < -4r + 10 part, multiply both sides of the inequality by -1, remembering to switch the sign, to get 6 > 4r – 10. Add 13 to each side to get 19 > 4r + 3. Then solve the right side of the inequality. Again, multiply both sides of the inequality by -1, switching the sign to get 4r – 10 ≥ -2. Now add 13 to each side of the equation: 4r + 3 ≥ 11. Finally, combine the equations to get the range for 4r + 3. Since the question asks for the least possible value of the expression, 11, (D), is the correct answer to the question. If you see the answer before the last step above, you don’t need to combine the equations.

If |x| = |2x – 1|, either x = 2x – 1 or -x = 2x – 1. The solutions to these equations are 1 and ​\( \frac{1}{3} \)​, respectively. However, the only thing you need to recognize is that the equation has two different solutions to establish that the answer is (C).

Plug in the answers. If you start with (B), the length is 8, and the width is half that, or 4. Area is length × width. The area of this rectangle is 8 × 4, which is nowhere near 128. Eliminate (A) and (B), as both are too small. Try (C): If the length is 16, the width is 8. So, does 128 = 16 × 8? You could write it all out, since you can’t use your calculator, but you can also estimate. 16 × 10 = 160, so 16 × 8 would be about 130. The number in (D) is too large and will give a weird fraction, so (C) is correct. Alternatively, write an equation. The equation is area = w × 2w. So, 128 = 2w². Divide by 2 to get 64 = w². Take the square root of both sides to find w = 8. The length is twice this width, so length = 2 × 8 = 16, and the answer is (C).

A question with unknown variables indicates a good place to plug in. You need numbers for x and y that will give you a negative product. Try x = 1 and y = -2. If you plug these into the statements in the Roman numerals, you find that (I) is false, but (II) and (III) are true. You can eliminate any answer choice that contains (I). This leaves (B) and (D). Now try different numbers to see if you can eliminate another choice. If you try x = -1 and y = 2, you find that (II) is false and (III) is still true. This leaves you with (B) as the only correct answer.