GRE Inequalities Questions Practice Questions with Answers

Concept Explanation

GRE inequalities questions are mathematical problems that require comparing two expressions using symbols like $<$, $>$, $\leq$, or $\geq$. These questions test your ability to manipulate algebraic statements while maintaining the correct relationship between values. Unlike standard equations where you solve for a specific point, inequalities result in a range of possible solutions. A fundamental rule to remember is that multiplying or dividing both sides of an inequality by a negative number reverses the direction of the inequality sign. For instance, if $-2x < 10$, then dividing by $-2$ results in $x > -5$.

Success on the Quantitative Reasoning section involves identifying when a variable's value is restricted by its sign or its relationship to other numbers. You will frequently encounter these concepts in GRE Prep materials, particularly in Quantitative Comparison and Multiple-Choice formats. Many problems involve absolute values, which create double-sided inequalities. For example, $|x| < 5$ translates to $-5 < x < 5$. Understanding these boundaries is critical for avoiding common traps set by the test makers at ETS.

Solved Examples

1. Linear Inequality: Solve for $x$ in the inequality $3x - 7 > 5x + 3$.
   1. Subtract $3x$ from both sides: $-7 > 2x + 3$.
   2. Subtract $3$ from both sides: $-10 > 2x$.
   3. Divide by $2$: $-5 > x$, or $x < -5$.
2. Inequality with Negative Multiplication: Solve for $y$ in $\frac{-y}{4} + 2 \leq 5$.
   1. Subtract $2$ from both sides: $\frac{-y}{4} \leq 3$.
   2. Multiply both sides by $-4$. Because we are multiplying by a negative number, we must flip the sign: $y \geq -12$.
3. Absolute Value Inequality: Find the range of $z$ if $|2z - 4| \leq 10$.
   1. Set up the compound inequality: $-10 \leq 2z - 4 \leq 10$.
   2. Add $4$ to all parts: $-6 \leq 2z \leq 14$.
   3. Divide all parts by $2$: $-3 \leq z \leq 7$.
4. Quadratic Inequality: Solve $x^2 - 5x + 6 < 0$.
   1. Factor the quadratic: $(x - 2)(x - 3) < 0$.
   2. Identify the critical points where the expression equals zero: $x = 2$ and $x = 3$.
   3. Test intervals: For $x < 2$, the product is positive. For $2 < x < 3$, the product is negative. For $x > 3$, the product is positive.
   4. The solution is $2 < x < 3$.

Practice Questions

1. If $4 - 2x \leq 12$, what is the smallest possible value for $x$?

2. Given that $a > b$ and $c < 0$, which of the following must be true? $ac < bc$ or $ac > bc$?

3. Solve the inequality: $|x + 5| > 8$.

4. If $-3 < y < 2$, what is the range of values for $y^2$?

5. Quantity A: $x^2$; Quantity B: $x^3$. Given that $0 < x < 1$, which quantity is greater?

6. Solve for $w$: $\frac{2w - 3}{5} \geq w + 1$.

7. If $x$ and $y$ are integers such that $2 < x < 5$ and $-4 < y < -1$, what is the maximum value of $x - y$?

8. Solve the inequality $x^2 - 9 \leq 0$.

9. If $\frac{1}{x} < \frac{1}{3}$ and $x > 0$, what can be concluded about $x$?

10. Quantity A: $|a + b|$; Quantity B: $|a| + |b|$. Compare the two quantities for any real numbers $a$ and $b$.

Answers & Explanations

1. Answer: -4. Subtract 4 from both sides to get $-2x \leq 8$. Divide by -2 and flip the sign: $x \geq -4$. The smallest value is -4.
2. Answer: $ac < bc$. When you multiply both sides of an inequality by a negative number ($c < 0$), the inequality sign must flip.
3. Answer: $x > 3$ or $x < -13$. This splits into $x + 5 > 8$ (which gives $x > 3$) and $x + 5 < -8$ (which gives $x < -13$).
4. Answer: $0 \leq y^2 < 9$. Squaring a range that includes zero means the minimum is 0. The maximum is the square of the endpoint with the largest absolute value, which is $(-3)^2 = 9$.
5. Answer: Quantity A. For fractions between 0 and 1, higher powers result in smaller values. For example, if $x = 0.5$, $x^2 = 0.25$ and $x^3 = 0.125$.
6. Answer: $w \leq - \frac{8}{3}$. Multiply by 5: $2w - 3 \geq 5w + 5$. Subtract $2w$: $-3 \geq 3w + 5$. Subtract 5: $-8 \geq 3w$. Divide by 3: $w \leq - \frac{8}{3}$.
7. Answer: 7. To maximize $x - y$, pick the largest $x$ (4) and the smallest $y$ (-3). $4 - (-3) = 7$.
8. Answer: $-3 \leq x \leq 3$. Factoring gives $(x-3)(x+3) \leq 0$. Testing intervals shows the expression is non-positive between -3 and 3 inclusive.
9. Answer: $x > 3$. Since $x$ is positive, we can multiply both sides by $3x$ without flipping the sign, getting $3 < x$.
10. Answer: Quantity B is greater than or equal to Quantity A. This is the Triangle Inequality theorem. If $a$ and $b$ have different signs, Quantity B will be strictly greater.

Frequently Asked Questions

When do I flip the inequality sign?

The inequality sign must be flipped whenever you multiply or divide both sides of the expression by a negative number. It does not flip when you add or subtract negative numbers.

How do I handle absolute values in inequalities?

Split the inequality into two separate cases: one where the expression inside is positive and one where it is negative. For $|x| < a$, use $-a < x < a$; for $|x| > a$, use $x > a$ or $x < -a$.

Can I multiply an inequality by a variable?

You should avoid multiplying by a variable unless you are certain of its sign. If the variable could be negative, you wouldn't know whether to flip the sign, which is a common source of errors on the GRE.

What is the difference between $<$ and $\leq$?

The symbol $<$ means "strictly less than," excluding the endpoint, while $\leq$ means "less than or equal to," including the endpoint in the solution set. This distinction is vital when performing a Retrieval Challenge during your study sessions.

How are inequalities tested in Quantitative Comparison questions?

These questions often ask you to determine if one quantity is always greater than the other. You must test various numbers, including fractions, zero, and negative integers, to see if the relationship holds true in all cases.