🔢 Quant

Inequalities & Absolute Value

The sign-flip rule, compound inequalities, sign analysis, and absolute value as distance.

~4h

to master

Inequalities: one rule above all

Solve linear inequalities exactly like equations — isolate the variable — with a single exception that decides almost every trap in this topic:

📐Core Rule

Multiplying or dividing both sides by a negative number flips the inequality sign. $6 > 2$ , but $-6 < -2$ . Adding/subtracting anything, or scaling by a positive number, leaves the direction alone.

The corollary the GMAT loves: you cannot multiply by a variable unless you know its sign. Given $\frac{a}{b} > 1$ , concluding $a > b$ is wrong — if $b < 0$ the flip applies ( $a = -3, b = -2$ satisfies the first, not the second). When a variable's sign is unknown, split into cases or test numbers from both sides of zero.

✏️Worked Example

Solve $7 - 2x < 1$ . Subtract 7: $-2x < -6$ . Divide by $-2$ and flip: $x > 3$ . Sanity-check with $x = 4$ : $7 - 8 = -1 < 1$ ✓.

Compound inequalities like

-3 \le 2x + 1 < 9

are two constraints sharing a middle: operate on all three parts at once →

-2 \le x < 4

. For integer-counting questions, list endpoints carefully: integers here are

-2, -1, 0, 1, 2, 3

— six values. Inclusive vs exclusive is half the question; drill it on .

Sign analysis for products and quotients

$\,(x-2)(x+5) > 0$ asks: where do the factors agree in sign? Mark the zeros ( $2$ and $-5$ ) on a number line; they split it into three zones; test one value per zone. Same machinery for quotients — but zeros of the denominator are excluded always.

Even powers are $\ge 0$ no matter what; $x^2 > 0$ fails only at $x = 0$ .
If $0 < x < 1$ : then $x^2 < x < \sqrt{x}$ — powers shrink proper fractions. Trap options regularly assume "squaring makes bigger".

Absolute value: think distance

$|x|$ is the distance from $x$ to zero — never negative, and $|{-3}| = |3| = 3$ .

📐Core Rule

The two unpackings (for $k > 0$ ): $|x| = k \;\Rightarrow\; x = k$ or $x = -k$ (two points) $|x| < k \;\Rightarrow\; -k < x < k$ (a strip) $|x| > k \;\Rightarrow\; x > k$ or $x < -k$ (two rays)

Shifted versions read as distance from a centre:

|x - 5| < 3

means "within 3 of 5", i.e.

2 < x < 8

. The centre is the midpoint of the answer band — often you can write the answer without algebra. Count solutions on .

⚠️GMAT Trap

Always check candidate solutions of absolute-value equations back in the original. Squaring or unpacking can introduce phantom solutions, especially when the variable also appears outside the bars ( $|x - 1| = 2x$ → only candidates with $2x \ge 0$ can survive).

⚡Shortcut

For "how many integers satisfy…" questions, solve the inequality once, then count with endpoints: an inclusive integer range $[a, b]$ holds $b - a + 1$ integers.

Checklist

Flip the sign on every negative multiply/divide
Unknown sign? No multiplying by the variable — case-split or test
Compound: operate on all three parts together
$|x - c|$ = distance from $c$ ; convert to a band or two rays
Verify abs-value equation roots in the original

Sample Questions

22 practice questions

Medium

How many integers $x$ satisfy both $-5 < 2x - 1 \le 7$ and $x \ne 1$ ?

Medium

How many integers $n$ satisfy $|n - 3| < 5$ ?

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