Inequalities Absolute Practice Questions

Master GMAT Inequalities Absolute with comprehensive practice questions. Build your quantitative reasoning skills through detailed explanations and strategic practice.

3 questions to try, free — no account needed

Question 1 of 5 Medium

If $ 3|3 - x| = 7 $, what is the product of all the possible values of $ x $?

View Explanation

Correct Answer: E

When solving an absolute value equation, it helps to first isolate the absolute value expression:

$ 3|3 - x| = 7 $
$ |3 - x| = \frac{7}{3} $

When removing the absolute value bars, we need to keep in mind that the expression inside the absolute value bars $ (3 - x) $ could be positive or negative. Let's consider both possibilities:

When $ (3 - x) $ is positive:
$ (3 - x) = \frac{7}{3} $
$ 3 - \frac{7}{3} = x $
$ \frac{9}{3} - \frac{7}{3} = x $
$ x = \frac{2}{3} $

When $ (3 - x) $ is negative:
$ -(3 - x) = \frac{7}{3} $
$ x - 3 = \frac{7}{3} $
$ x = \frac{7}{3} + 3 $
$ x = \frac{7}{3} + \frac{9}{3} $
$ x = \frac{16}{3} $
So, the two possible values for $ x $ are $ \frac{2}{3} $ and $ \frac{16}{3} $. The product of these values is $ \frac{32}{9} $.
The correct answer is E.

Question 2 of 5 Medium

If $ |a| = \frac{1}{3} $ and $ |b| = \frac{2}{3} $, which of the following CANNOT be the result of $ a + b $?

View Explanation

Correct Answer: D

The possible values for $ a $ and $ b $ are $ \pm \frac{1}{3} $ and $ \pm \frac{2}{3} $ respectively.
We can list the possible sums:

$$\begin{array}{|c|c|c|} \hline a & b & a+b \\ \hline \frac{1}{3} & \frac{2}{3} & 1 \\ \hline \frac{1}{3} & -\frac{2}{3} & -\frac{1}{3} \\ \hline -\frac{1}{3} & \frac{2}{3} & \frac{1}{3} \\ \hline -\frac{1}{3} & -\frac{2}{3} & -1 \\ \hline \end{array}$$

Comparing with the options, $ \frac{2}{3} $ is NOT a possible sum.
The correct answer is D.

Question 3 of 5 Medium

If $ \sqrt{(x + 4)^2} = 3 $, which of the following could be the value of $ x - 4 $?

View Explanation

Correct Answer: A

Simplify the equation: $ \sqrt{(x + 4)^2} = |x + 4| = 3 $.

This yields two cases:
$ x + 4 = 3 \implies x = -1 $
$ x + 4 = -3 \implies x = -7 $

The question asks for the value of $ x - 4 $, not $ x $ itself. Calculate $ x - 4 $ for both solutions:
If $ x = -1 $: $ x - 4 = -1 - 4 = -5 $ (not among the answer choices).
If $ x = -7 $: $ x - 4 = -7 - 4 = -11 $.

Since $ -11 $ is the only value that appears among the answer choices, the correct answer is A.

Question 4 of 5 Medium

If $ x > y $, $ x^2 - 2xy + y^2 - 9 = 0 $, and $ x + y = 15 $, what is $ x $?

Sign up to take the full adaptive mock

These 3 questions are a preview. The real test adapts to your level across all three GMAT sections.

Question 5 of 5 Medium

If $ b < c < d $ and $ c > 0 $, which of the following cannot be true if $ b, c $ and $ d $ are integers?

Sign up to take the full adaptive mock

These 3 questions are a preview. The real test adapts to your level across all three GMAT sections.