GRE Solutions Manual, Problem 5.5

Exam Prep, GRE, Quantitative Reasoning

This page is part of my unofficial solutions manual to the GRE Paper Practice Book (2e), a free resource available on the ETS website. They publish the questions; I explain the answers. If you haven’t worked through the Practice Book, give Section 5 a shot before reading this!

5.5: Inequalities, Part 1

This problem is a fairly straightforward check on your knowledge of inequalities. (For a more advanced problem on the same theme, see 5.15.) Just like an equation, an inequality can be manipulated via addition, subtraction, multiplication, or division — as long as you apply the same operation to both sides. There’s one important difference, however:

When you multiply (or divide) both sides of an inequality by a negative number, you have to flip the signs.

Equivalently, we could say that when you multiply both sides of an inequality by a negative number, the direction of the inequality changes. For example, the statement

Three is less than five.

means that 3 is to the left of 5 on the number line. But if we multiply both sides by -1, we get:

The quantity "negative 1 times 3" is greater than the quantity "negative 1 times 5."
Negative three is greater than negative five.

On the number line, -3 is to the right of -5.

Now let’s turn back to problem 5.5. We need to find out whether t is greater than 0, equal to 0, less than 0, or whether this relationship changes for different values of t. The clue is in that other variable, r.

Looking at the leftmost term of the inequality, we can see that the product rt is negative. This tells us two things:

  • Neither r nor t can be zero, since then the product would be zero. (This lets us rule out answer C.)
  • Either r or t, but not both, must be negative. (If both were negative, their product would be positive.)

Fortunately, we get some additional information about r in the third term of the inequality:

Zero is less than negative r.

If -r is positive, then r itself must be negative. This means that t cannot be negative, and we already determined that t can’t be zero. So t must be positive: quantity A (t) is greater than quantity B (0), and the correct answer is (A).


Math Review Reference

For more on this topic, see the following section of the GRE Math Review:

  • 1.5: Real Numbers (pp. 7-8)