Inequalities and absolute values are high-yield GRE Quant topics that test rigorous logical deduction, sign awareness, and range reasoning. Master the core algebraic rules, distance mechanics, quadratic shortcuts, and high-frequency Quantitative Comparison traps.
Master the rules for addition, multiplication, and sign flips. Understand why dividing by an unknown variable is the #1 mistake on the GRE and how to factor instead.
Unlike equations, inequalities carry direction. Performing algebraic operations—especially multiplication, division, and subtraction—requires strict adherence to sign rules.
You can add or subtract any number (positive or negative) to both sides without changing the direction of the inequality:
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 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 inequalities, doing this is FATAL unless you already know for certain whether the variable is positive or negative.
Solve for $x$: $$ x^2 < 4x $$
The Incorrect Temptation: Blindly divide both sides by $x$ to conclude $x < 4$.
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 suggests that numbers like $x = -2$ or $x = 0$ are solutions:
Dividing by $x$ completely erased the lower boundary ($x > 0$) and generated a false solution set!
The Correct First-Principles Method: Never divide by an unknown variable! Subtract $4x$ to set one side to zero and factor:
The real solution is strictly $0 < x < 4$. Golden 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.
Because the absolute value $|A|$ denotes distance from zero, it is always non-negative. Solving absolute value inequalities requires splitting them into exactly two straightforward cases depending on whether the sign 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 $+B$:
Standard 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 continuous open segment between $-12$ and $-4$.
When an absolute value expression is greater than a positive bound $B$, the expression splits into two separate outside wings:
Standard 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 rays pointing outwards 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
Solve quadratic inequalities without endless sign charts. Learn the roots test and the standard Between vs Outside shortcut.
Drawing sign charts or testing multiple test points on a number line is slow and prone to errors. For any factored quadratic expression with a positive $x^2$ coefficient, the sign of the product changes only at its 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: $x^2 - 5x + 6 < 0 \implies 2 < x < 3$.
Manipulate roots r₁ and r₂ dynamically. Toggle between negative (< 0) and positive (> 0) to visualize parabola regions and critical sign changes.
Between Roots: 2 < x < 5
When a factored quadratic is < 0 (with positive x²), the solution is strictly trapped BETWEEN the roots:
Examine how numbers between 0 and 1 break normal intuition. Master the power inversion hierarchy: x³ < x² < x < √x < 1 < 1/x.
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 GRE, 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:
In Quantitative Comparison questions without square roots, testing $x = \frac{1}{2}$ is the fastest scratchpad check:
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: 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 product sign traps. Apply the proven 5-number testing checklist: 2, 1/2, 0, -1/2, -2.
Quantitative Comparison Problem:
Given: $x < 3$
Students mistakenly square both sides: $x < 3 \implies x^2 < 9$, concluding Quantity B is greater.
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