7. B To solve this equation, use cross multiplication to get (2x)(x + 2) = (x^2 + 1)(2). Expand the
equation to get 2x^2 + 4x = 2x^2 + 2. Once you combine like terms, the result is 2x^2 – 2x^2 + 4x = 2
or 4x = 2. Solve for x by dividing both sides by 4 to get x = , which is (B).
A 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 =
2 y^2 , 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 states.
D 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.
C If |x| = |2x − 1|, either x = 2x − 1 or –x = 2x − 1. The solutions to these equations are 1 and ,
respectively. However, the only thing you need to recognize is that the equation has two
different solutions to establish that the correct answer is (C).- D This is a system of equations question in disguise. First, locate a piece of information in this
question that you can work with. “The sum of three numbers, a, b, and c, is 400,” seems very
straightforward. Write the equation a + b + c = 400. Now the question tells you that “one of the
numbers, a, is 40 percent less than the sum of b and c.” Translate this piece by piece to get a =
(1 – 0.4) (b + c), or a = 0.6(b + c). Distribute the 0.6 to get a = 0.6b + 0.6c. Arrange these
variables so they line up with those in the first equation as a – 0.6b – 0.6c = 0. To solve for b +
c, stack the equations and multiply the second equation by –1:
a + b + c = 400
–1(a – 0.6b – 0.6c) = 0(–1)Now solve:Simplify by dividing both sides by 1.6 to get b + c = 250. The correct answer is (D).