"If |x| = |y| and xy = 0, Which of the Following Must Be True?" — GMAT® Worked Solution

From Episode 45 of Real GMAT® Problems (The GMAT® Strategy Podcast). For the full strategy behind plugging in numbers, read: GMAT® Algebra: What to Do When You're Stuck.

The Problem

Source: Official Guide for GMAT® Review, 11th Edition

If |x| = |y| and xy = 0, which of the following must be true?

(A) xy² > 0
(B) x²y > 0
(C) x + y = 0
(D) x / (y + 1) = 2
(E) 1/x + 1/y = 1/2

Try it before reading on.

Why This Problem Trips People Up

This looks like a standard algebra question at first glance. You see absolute values and equations and your instinct says: solve it.

But there's no single variable to isolate here. The problem gives you two conditions and five answer choices. It asks which one MUST be true — meaning it has to hold 100% of the time, as long as the conditions are met.

That "must be true" language is one of the strongest signals that plugging in numbers can work well.

Setting Up Columns

Make two columns on your scratch paper. Write x at the top of one and y at the top of the other.

This sounds basic. It prevents the most common mistake: forgetting what you plugged in for which variable as you work through five answer choices.

Trying Some Numbers

Start simple. Try x = 2 and y = 2.

Does |x| = |y|? Yes. |2| = |2|.

Does xy = 0? No. 2 × 2 = 4.

So that pair doesn't work. But it's not wasted — it tells you something. Two non-zero numbers with the same absolute value will never multiply to zero.

Try x = −2 and y = 2.

Does |x| = |y|? Yes. |−2| = |2|.

Does xy = 0? No. −2 × 2 = −4.

Still no. Positive, negative — doesn't matter. If both numbers are non-zero, the product can't be zero.

That realization pushes you in the right direction. For xy to equal zero, at least one of the variables has to be zero. And if one is zero, the absolute value condition forces the other to be zero too.

So x = 0 and y = 0.

Testing the Answer Choices

Now plug x = 0 and y = 0 into each option.

(A) xy² > 0 → 0 × 0² = 0. Zero is not greater than zero. Eliminate.

(B) x²y > 0 → 0² × 0 = 0. Same situation. Eliminate.

(C) x + y = 0 → 0 + 0 = 0. That works.

(D) x / (y + 1) = 2 → 0 / (0 + 1) = 0/1 = 0. That's not 2. Eliminate.

(E) 1/x + 1/y = 1/2 → 1/0 + 1/0. Division by zero isn't allowed on the GMAT®. Any answer choice that requires dividing by zero can't be correct. Eliminate.

That leaves (C).

The answer is (C).

What to Take Away

Three things worth noting here.

"Must be true" questions and plugging in numbers go together. You don't need to prove the answer algebraically. You need to eliminate the four that fail. One set of valid numbers can wipe out most of the wrong answers.

Numbers that don't satisfy the constraints are still useful. Trying x = 2 and y = 2 didn't "work" — but it revealed that both variables have to be zero. When you're stuck, knowing what doesn't work often leads you to what does.

Watch for division by zero in the answer choices. If an answer choice would require dividing by zero with valid inputs, you can eliminate it right away.

Ready for a harder one? Same approach, but the algebra is much messier and the payoff for plugging in is even bigger: "(x + 1)/(x − 1), Squared" — replace x with 1/x.

Want the full strategy? Read: GMAT® Algebra: What to Do When You're Stuck

From Episode 45 of Real GMAT® Problems (The GMAT® Strategy Podcast).