GMAT Quantitative: Inequalities and Absolute Value

Mastering inequalities and absolute value is non-negotiable for a high GMAT Quantitative score. These concepts are the bedrock for solving data sufficiency and problem-solving questions involving ranges, distances, and logical constraints. Your ability to manipulate these expressions accurately directly translates to stronger performance in algebra, word problems, and advanced reasoning—skills essential for the data-driven decision-making of an MBA.

Foundational Rules and Number Line Logic

Before tackling complex problems, you must internalize the core rules of inequality manipulation. You can add or subtract any number from both sides without changing the inequality's direction. However, multiplying or dividing by a negative number flips the inequality sign. For example, if $-2x<8$, dividing by $-2$ gives $x>-4$. Forgetting to flip the sign is a classic error.

The most powerful visual tool is the number line approach. It transforms abstract inequalities into a clear spatial representation. To solve a linear inequality like $3x+7\ge 1$, first isolate $x$ to get $x\ge -2$. On a number line, this is a solid circle at $-2$ with a shaded ray extending to the right. For compound statements like "$x<1$ OR $x\ge 4$," you shade the two separate, non-overlapping regions. This visualization is crucial for avoiding logic errors in data sufficiency questions asking for a range of possible values.

Solving Compound and Quadratic Inequalities

Compound inequalities often appear in the form $a<bx+c<d$. The key is to perform operations on all three parts simultaneously. To solve $-3<2x-5\le 7$, add 5 to each part: $2<2x\le 12$. Then divide each part by 2: $1<x\le 6$. The solution is all numbers between 1 and 6, excluding 1 but including 6. On a number line, this is a line segment from an open circle at 1 to a closed circle at 6.

For quadratic inequalities (e.g., $x^2-x-6>0$), the "testing regions" method is reliable. First, solve the related equation $x^2-x-6=0$, which factors to $(x-3)(x+2)=0$, giving critical points $x=-2$ and $x=3$. Plot these on a number line, creating three regions: $x<-2$, $-2<x<3$, and $x>3$. Now test a number from each region in the original inequality. For $x<-2$, test $x=-3$: $(-3{)}^2-(-3)-6=6>0$ (True). For $-2<x<3$, test $x=0$: $0-0-6=-6>0$ (False). For $x>3$, test $x=4$: $16-4-6=6>0$ (True). Since the inequality is "> 0," the solution is the regions that yielded "True": $x<-2$ OR $x>3$.

Absolute Value as Distance

The absolute value of a number, denoted $|x|$, represents its distance from zero on the number line, which is always non-negative. This "distance" interpretation is the key to solving all absolute value equations and inequalities. The equation $|x|=5$ asks, "Which numbers are 5 units from zero?" The answer is $x=5$ or $x=-5$.

This extends to expressions: $|A|=B$ (where $B\ge 0$) means $A=B$ OR $A=-B$. To solve $|2x-1|=7$, you create two linear equations: $2x-1=7$ and $2x-1=-7$, solving to get $x=4$ or $x=-3$. Always check that your solutions satisfy the original equation, though they typically will if your algebra is correct.

Solving Absolute Value Inequalities

The distance concept elegantly frames inequalities. The statement $|A|<B$ (with $B>0$) means "the distance of A from zero is less than B." This translates to the compound inequality: $-B<A<B$. It's a single "AND" condition. For example, $|x+3|<4$ becomes $-4<x+3<4$, which solves to $-7<x<1$.

Conversely, $|A|>B$ means "the distance of A from zero is greater than B." This is an "OR" situation: $A<-B$ OR $A>B$. To solve $|2x-5|\ge 3$, set up: $2x-5\le -3$ OR $2x-5\ge 3$. Solving these gives $x\le 1$ OR $x\ge 4$. A common GMAT trap is to incorrectly combine this into a single compound inequality; these are two distinct, non-adjacent ranges. On a number line, you would have two shaded rays heading left from 1 and right from 4.

Common Pitfalls

1. Failing to Flip the Inequality Sign with Negatives: This is the most frequent algebraic error. Whenever you multiply or divide an inequality by a negative number, the sign must reverse. Correction: Isolate the variable carefully. If you divide by a negative, immediately reverse the sign before performing the arithmetic.

2. Misrepresenting "Greater Than" Absolute Value Inequalities: A solution to $|x|>5$ is $x>5$ OR $x<-5$. Writing it as $x>5$ AND $x<-5$ is impossible (no number satisfies both) and misses half the solution. Correction: Remember the "OR" structure for ">" and the "AND" structure for "<". Use the number line to visualize the two separate regions.

3. Ignoring the Non-Negative Constraint in Absolute Value Equations: If you derive an equation like $|x|=-3$, stop immediately. Distance cannot be negative. Correction: Recognize that such an equation has no solution. In more complex problems, this might arise from an intermediate step, signaling that a potential solution path is invalid.

4. Improper Handling of Zero in Denominators within Inequalities: If you have an inequality like $\frac{1}{x}>2$, you cannot simply cross-multiply by $x$ because its sign is unknown. If $x$ is negative, the inequality sign would flip. Correction: Bring all terms to one side to get $\frac{1}{x}-2>0$, combine into a single fraction, and then use the testing regions method on the critical points, which include where the denominator is zero ($x=0$).

Summary

• Inequality manipulation requires flipping the sign when multiplying or dividing by a negative number. The number line is an indispensable tool for visualizing solutions.
• Solve compound inequalities by performing operations on all three parts simultaneously. Solve quadratic inequalities by finding critical points and testing regions.
• Interpret absolute value as distance. For equations: $|A|=B$ means $A=B$ OR $A=-B$. For inequalities: $|A|<B$ means $-B<A<B$ ("AND"); $|A|>B$ means $A<-B$ OR $A>B$ ("OR").
• Always be vigilant for common traps, especially sign-flipping errors, misusing "AND"/"OR" for absolute value, and undefined expressions involving zero or negative denominators.