Inequalities and absolute values are high-frequency GMAT Quant topics tested extensively in Problem Solving and Data Sufficiency. Master the core algebraic rules, distance mechanics, quadratic shortcuts, and high-yield DS trap strategies.
Master the rules for addition, multiplication, and sign flips. Understand why dividing by an unknown variable is the #1 trap in GMAT Data Sufficiency and how to factor instead.
In GMAT Quantitative Reasoning, inequalities are tested most aggressively in Data Sufficiency (DS). Unlike standard equations, inequalities carry direction, making sign awareness and algebraic operations the primary test ground for 700+ level traps.
You can add or subtract any number (positive or negative) to both sides of an inequality without changing its direction:
If $a < b$, then $a + c < b + c$
If $a < b$, then $a - c < b - c$
Example: $3 < 7 \implies 3 - 9 < 7 - 9 \implies -6 < -2$. The relationship remains completely true.
Multiplying or dividing by a positive number preserves direction. Multiplying or dividing by a negative number flips the inequality sign:
Positive: $3 < 7 \implies 3(4) < 7(4) \implies 12 < 28$
Negative: $3 < 7 \implies 3(-1) > 7(-1) \implies -3 > -7$
Whenever you multiply or divide both sides by a negative value, you must immediately reverse the inequality arrow.
In standard algebra ($ab = c$), you can divide by $b$ to get $a = c/b$. In GMAT inequalities, doing this is FATAL unless the problem explicitly states that $b > 0$!
Question: Is $x < 4$?
Statement (1): $x^2 < 4x$
The Incorrect Temptation: Divide both sides by $x$ to conclude $x < 4$, picking Statement 1 as Sufficient.
Why this is wrong: When you divide by $x$, you implicitly assume $x$ is positive without stating the condition $x > 0$. The resulting answer $x < 4$ mistakenly includes invalid values like $x = -2$ or $x = 0$:
Dividing by $x$ completely erased the lower boundary ($x > 0$) and produced a distorted solution set!
The Correct GMAT First-Principles Method: Never divide by an unknown variable! Subtract $4x$ to set one side to zero and factor:
Since $0 < x < 4$, $x$ is strictly between 0 and 4, which means $x$ is always less than 4! The statement is indeed Sufficient, but understanding that the real range is $0 < x < 4$ protects you when the question asks "Is $x > 0$?". Golden GMAT Rule: Subtract and factor; never divide by an unknown variable.
If the inequality signs point in the same direction, you can add them column by column:
Subtracting inequalities facing the same direction produces invalid conclusions. (e.g. $10 < 20$ and $2 < 15$; subtracting gives $8 < 5$, which is FALSE!). To subtract, multiply the second inequality by $-1$, flip its sign, and add.
Master the standard algebraic split: trap less-than between -B and +B, and separate greater-than into two disjoint wings for GMAT PS and DS.
Because the absolute value $|A|$ represents distance from zero on a number line, it is always non-negative. In the GMAT, absolute value expressions must be solved by strictly splitting them into two algebraic cases based on whether the relation is less than or greater than.
When an absolute value expression is less than a positive bound $B$, the expression is trapped between $-B$ and $+B$:
GMAT PS Example: Solve $|x + 8| < 4$
-4 < x + 8 < 4
-4 - 8 < x < 4 - 8 (subtract 8 from all three parts)
-12 < x < -4
On the number line, the solution is the single continuous interval between $-12$ and $-4$.
When an absolute value expression is greater than a positive bound $B$, the expression splits into two separate outside wings:
GMAT DS Example: Solve $|4x + 8| > 20$
4x + 8 < -20 OR 4x + 8 > 20
4x < -28 OR 4x > 12
x < -7 OR x > 3
On the number line, the solution splits into two separate outside rays pointing to $\pm\infty$.
Manipulate |x + k| < B and |x + k| > B. Follow the step-by-step algebraic solution and view the resulting solution interval shaded on the number line.
Single continuous interval: (-12, -4)
Step 1 (Split): Trap expression between -4 and +4:
-4 < x + 8 < 4
Step 2 (Isolate x): Subtract 8 from all parts:
-12 < x < -4
Prerequisite: Review quadratic factoring and roots. Solve quadratic inequalities quickly using root tests and the standard Between vs Outside shortcut.
Drawing complex sign charts or testing multiple test points on a number line is a major time sink in GMAT Quant. For any factored quadratic expression with a positive $x^2$ coefficient, the sign of the product changes only at its roots.
Before tackling quadratic inequalities, make sure you are confident factoring quadratics and finding roots.
Consider the factored quadratic $(x - 2)(x - 3)$. The roots are $2$ and $3$. Let us examine the product across all three number-line intervals:
| Interval | Sign of $(x - 2)$ | Sign of $(x - 3)$ | Product $(x - 2)(x - 3)$ | Conclusion |
|---|---|---|---|---|
| $x > 3$ | + | + | POSITIVE (> 0) | Both factors positive |
| $2 < x < 3$ | + | - | NEGATIVE (< 0) | Opposite signs create negative! |
| $x < 2$ | - | - | POSITIVE (> 0) | Both factors negative |
If the quadratic expression is negative, the values of $x$ are strictly trapped between the two roots.
If the quadratic expression is positive, the values of $x$ lie strictly outside the two roots (less than the smaller root or greater than the larger root).
Always ensure the coefficient of $x^2$ is positive before applying this shortcut. If given $-x^2 + 5x - 6 > 0$, multiply the entire inequality by $-1$ and flip the sign: $x^2 - 5x + 6 < 0 \implies 2 < x < 3$.
Examine how numbers between 0 and 1 break normal intuition. Master the power inversion hierarchy: x³ < x² < x < √x < 1 < 1/x and higher powers for negatives.
In everyday life, our intuition is calibrated to positive integers ($x > 1$), where multiplying makes numbers bigger and taking roots makes them smaller. In the GMAT, questions deliberately target the Fractional Zone ($0 < x < 1$), where this intuition completely reverses.
Let $x = \frac{1}{4}$ (a clean perfect-square fraction). Notice how comparing fractions with equal numerators ($1$) makes the ordering instant—larger denominators mean smaller numbers:
On the GMAT (no calculator), if a question does not test roots, testing $x = \frac{1}{2}$ on your scratch paper is the fastest mental model:
Comparing denominators ($8 > 4 > 2$) immediately proves higher powers shrink towards zero, while the reciprocal expands past $1$!
| Regime | Test Number | Relative Ordering of Powers | Key Takeaway & Exponent Rules |
|---|---|---|---|
| $x > 1$ | $x = 2$ | $\frac{1}{x} < 1 < x < x^2 < x^3$ | Standard intuition holds: higher powers grow larger, reciprocal shrinks. |
| $0 < x < 1$ | $x = \frac{1}{2}$ | $x^3 < x^2 < x < \sqrt{x} < \frac{1}{x}$ | Powers shrink; roots and reciprocals grow! |
| $-1 < x < 0$ | $x = -\frac{1}{2}$ | $\frac{1}{x} < x < x^3 < 0 < x^2$ | $x^2$ is positive; all odd powers and reciprocals are negative. |
| $x < -1$ | $x = -2$ | $x^3 < x < \frac{1}{x} < 0 < x^2$ | $x^2$ is positive; $x^3$ is most negative. Higher Powers Rule: Even powers grow more positive ($x^4 > x^2$; $16 > 4$), while odd powers plunge even further negative ($x^5 < x^3$; $-32 < -8$). |
Slide x through the fractional zone (0 < x < 1) and compare with negative and greater-than-1 regimes. Watch power and reciprocal rankings invert in real time.
Squaring shrinks a number (x² < x), while taking the reciprocal or square root expandsit (1/x > √x > x).
Strict Order: x³ < x² < x < 1/x
Avoid fatal squaring and variable cross-multiplication traps in DS. Apply the proven 5-number testing checklist: 2, 1/2, 0, -1/2, -2.
Data Sufficiency Scenario:
Is $x < 3$?
Solving $x^2 < 9$ using root shortcuts gives $-3 < x < 3$.
Now suppose the question asked: "Is $x > 0$?" given $x^2 > 9$.
Share this guide with your WhatsApp group or study buddy to compare problem-solving methods and benchmark your pace on the timed practice drills.
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