McGraw-Hill Education GRE 2019

Step 3: Combine the inequalities:

x < 4 and x > −10

↓

−10 < x < 4

Test Positives and Negatives

When answering a “must be true” question or Quantitative Comparison question

with absolute values, it is helpful to test positive and negative cases.

For this question, indicate all of the answer choices that apply.

If x ≠ 0, then which of the following must be true? (Indicate all that apply).

A^ |x| = x

B^ √x^2 = |x|

C^

|x|

x = 1

D^

|x|^2

|x| = x

E^ |x| × |x| = x^2

SOLUTION: Choose a positive and a negative value for x, and see which choices

are true for both cases. Let’s use −2 and 2 for x. Note that you will start with

the negative case, since this is the case most likely to contradict the given

equations.

A: |−2| = 2. 2 ≠ −2 → Eliminate Choice A.

B: √(–2)^2 = √ 4 = 2. |−2| = 2. The equation is true when x = −2.

Now try 2: √ 22 = √ 4 = 2. |2| = 2. The equation is true when x = −2.

→ Keep Choice B.

C: |2|–2 = –2^2 = −1. −1 ≠ 1 → Eliminate Choice C.

D: |–2|

2

|–2| =

22

2 =

4

2 = 2. 2 ≠ −2 → Eliminate Choice D.

E: |−2| × |−2| = 2 × 2 = 4. (−2)^2 = 4. The equation is true when x = −2.

Now try 2: |2| × |2| = 2 × 2 = 4. (2)^2 = 4. The equation is true when x = 2.

→ Keep Choice E.

The correct answer is B and E.

306 PART 4 ■ MATH REVIEW

03-GRE-Test-2018_173-312.indd 306 12/05/17 11:56 am