Inequalities and absolute values are where careful students lose points to careless moves. The rules are few, but each has an exception the GMAT likes to test.
With inequalities you can add or subtract anything on both sides, but multiplying or dividing by a negative number flips the sign, and you cannot multiply or divide by a variable unless you know its sign. Squaring is risky too: if x² is greater than y², you only know that |x| is greater than |y|, not that x is greater than y. Cubing, by contrast, preserves order for every real number.
For absolute values, think distance. |x − 3| is the distance between x and 3 on the number line, so an inequality such as |2x − 5| < 9 describes an interval, not two unrelated equations. When you do split into cases, solve each case and then check every candidate in the original equation. Extraneous solutions, values that fall out of the algebra but fail the original condition, are a favourite Data Sufficiency trap.
When counting integer solutions, watch the endpoints: a strict inequality excludes them, and the off-by-one answer is almost always among the choices. In Data Sufficiency, test a negative number, a fraction between 0 and 1, and zero before trusting a pattern.
4 practice questions
Data SufficiencyInequalitiesMedium
Question 1
Is $x > y$?
(1) $x^3 > y^3$
(2) $x^2 > y^2$
AStatement (1) ALONE is sufficient, but statement (2) ALONE is NOT sufficient.
BStatement (2) ALONE is sufficient, but statement (1) ALONE is NOT sufficient.
CBOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
DEACH statement ALONE is sufficient.
EStatements (1) and (2) TOGETHER are NOT sufficient.
Show answer & explanation
Correct answer: A
Key Concept or Approach
$t \mapsto t^3$ is strictly increasing on all real numbers, so it preserves order. $t \mapsto t^2$ does not, because it ignores sign.
Step-by-Step Solution
Step 1: Statement (1): the cube function is strictly increasing, so $x^3 > y^3$ implies $x > y$. Sufficient.
Step 2: Statement (2): $x^2 > y^2$ only means $|x| > |y|$. $x = 3, y = 1$ gives yes; $x = -3, y = 1$ gives no. Not sufficient.
Step 3: Only statement (1) is sufficient.
Final Answer
Answer: A
Common trap: D: students treat squaring like cubing and conclude $x^2 > y^2$ implies $x > y$.
Problem SolvingAbsolute Values/ModulesInequalitiesMedium-hard
Question 2
How many integers $x$ satisfy $|2x - 5| < 9$?
A$6$
B$7$
C$8$
D$9$
E$10$
Show answer & explanation
Correct answer: C
Key Concept or Approach
Convert the absolute-value inequality into a double inequality.
Common trap: 9 - counting 7 - (-2) = 9, or including an endpoint even though the inequality is strict.
Data SufficiencyAbsolute Values/ModulesAlgebraHard
Question 3
What is the value of $x$?
(1) $|x - 3| = 2x$
(2) $x^2 - 4x + 3 = 0$
AStatement (1) ALONE is sufficient, but statement (2) ALONE is NOT sufficient.
BStatement (2) ALONE is sufficient, but statement (1) ALONE is NOT sufficient.
CBOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
DEACH statement ALONE is sufficient.
EStatements (1) and (2) TOGETHER are NOT sufficient.
Show answer & explanation
Correct answer: A
Key Concept or Approach
When an absolute value equals an expression in $x$, split into cases and check each candidate, because the right side must be non-negative.
Step-by-Step Solution
Step 1: Statement (1): if $x \ge 3$, then $x - 3 = 2x$ gives $x = -3$, which contradicts $x \ge 3$. If $x < 3$, then $3 - x = 2x$ gives $x = 1$; check: $|1 - 3| = 2 = 2(1)$. The only solution is $x = 1$. Sufficient.
Step 2: Statement (2): $(x - 1)(x - 3) = 0$, so $x = 1$ or $x = 3$. Not sufficient.
Step 3: Only statement (1) is sufficient.
Final Answer
Answer: A
Common trap: C or E: students keep $x = -3$ as a second root of (1), think it is not sufficient, and then combine it with (2).
Data SufficiencyAbsolute Values/ModulesInequalitiesVery hard
Question 4
Is $|x| \le 1$?
(1) $x^4 < x^2$
(2) $|x + 1| + |x - 1| = 2$
AStatement (1) ALONE is sufficient, but statement (2) ALONE is NOT sufficient.
BStatement (2) ALONE is sufficient, but statement (1) ALONE is NOT sufficient.
CBOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
DEACH statement ALONE is sufficient.
EStatements (1) and (2) TOGETHER are NOT sufficient.
Show answer & explanation
Correct answer: D
Key Concept or Approach
$|x + 1| + |x - 1|$ is the sum of the distances from $x$ to $-1$ and to $1$. It equals 2 exactly when $x$ lies between them, and is larger outside.
Step-by-Step Solution
Step 1: Statement (1): $x^4 < x^2 \Rightarrow x^2(x^2 - 1) < 0$, so $x \ne 0$ and $x^2 < 1$. Hence $|x| < 1$, so $|x| \le 1$. Sufficient.
Step 2: Statement (2): for $-1 \le x \le 1$ the sum is $(x + 1) + (1 - x) = 2$. For $x > 1$ it is $2x > 2$; for $x < -1$ it is $-2x > 2$. So $-1 \le x \le 1$, and $|x| \le 1$. Sufficient.
Step 3: Each statement alone is sufficient.
Final Answer
Answer: D
Common trap: A: students think (2) has infinitely many solutions so it can't be sufficient, missing that every one of them satisfies $|x| \le 1$.
