Given that |a+b| < |a| + |b|, which of the fo… | GMAT Sprint

Free Practice Question

PS · Problem Solving

555-605

Absolute Values/Modules

Given that |a+b| < |a| + |b|, which of the following statements must hold true?
I. $ab < 0$
II. $ab^3 < 0$
III. $a + b < 0$

Answer Choices

Correct answer marked below

I only

II only

III only

I & II only

Correct

II & III only

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Explanation

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Official Explanation

Official

Starting with the inequality

|a+b| < |a| + |b|

, we can square both sides to obtain:

a^2 + 2ab + b^2 < a^2 + 2|ab| + b^2

, which simplifies to

2ab < 2|ab|

.

This implies that

ab < |ab|

, indicating that ab must be less than zero, meaning a and b have opposite signs.

Therefore, I.

ab < 0

is true.…

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Comments

RupinderJul 5, 2024

taking my GMAT next month, this helps!

SarasMar 17, 2025

Thankss

What this question tests

What it tests

Your ability to solve absolute-value equations and inequalities by considering both cases.

Common trap

Dropping the absolute value bars entirely or missing that |x| = k has two solutions when k > 0.

Given that |a+b| < |a| + |b|, which of the following statements must hold true? I. ab<0ab < 0 ab<0II. ab3<0ab^3 < 0 ab3<0III. a+b<0a + b < 0a+b<0