Inequalities & Absolute Value Cheat Sheet

A printable reference covering inequality symbols, graphing, interval notation, compound inequalities, and absolute value equations and inequalities for grades 8-10.

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Inequalities show when one quantity is greater than, less than, or not equal to another quantity. This cheat sheet helps students solve, graph, and write inequalities using symbols, number lines, and interval notation. It is especially useful when checking when an answer should include an endpoint or extend forever. Students in grades 8 through 10 use these skills in algebra, graphing, word problems, and functions. The most important rule is that multiplying or dividing both sides by a negative number reverses the inequality sign. Compound inequalities use the words and and or to describe overlapping or combined solution sets. Absolute value means distance from $0$ , so $|x|=a$ usually creates two possible equations when $a>0$ . Absolute value inequalities follow two main patterns: $|x|<a$ means $-a<x<a$ , while $|x|>a$ means $x<-a$ or $x>a$ .

Key Facts

The symbols $<$ , $>$ , $\leq$ , and $\geq$ mean less than, greater than, less than or equal to, and greater than or equal to.
When adding or subtracting the same value on both sides, the inequality direction stays the same, such as $x+3<7 \Rightarrow x<4$ .
When multiplying or dividing both sides by a negative number, reverse the inequality sign, such as $-2x<8 \Rightarrow x>-4$ .
Use an open circle for $<$ or $>$ and a closed circle for $\leq$ or $\geq$ on a number line.
A compound and inequality means both conditions must be true, such as $-2<x\leq5$ .
A compound or inequality means at least one condition must be true, such as $x<-3$ or $x\geq4$ .
For $a>0$ , the equation $|x|=a$ has two solutions: $x=a$ or $x=-a$ .
For $a>0$ , $|x|<a$ becomes $-a<x<a$ , and $|x|>a$ becomes $x<-a$ or $x>a$ .

Vocabulary

Inequality: An inequality is a mathematical statement that compares two expressions using symbols such as $<$ , $>$ , $\leq$ , or $\geq$ .
Solution set: A solution set is the collection of all values that make an equation or inequality true.
Interval notation: Interval notation describes a set of numbers using parentheses for excluded endpoints and brackets for included endpoints, such as $(2,5]$ .
Compound inequality: A compound inequality combines two inequalities using and or or, such as $x>1$ and $x<6$ .
Absolute value: Absolute value is the distance of a number from $0$ on a number line, written as $|x|$ .
Endpoint: An endpoint is a boundary value of an interval, and it is included when the symbol is $\leq$ or $\geq$ .

Common Mistakes to Avoid

Not flipping the inequality when multiplying or dividing by a negative; this is wrong because $-3x<12$ becomes $x>-4$ , not $x<-4$ .
Using a closed circle for $<$ or $>$ ; this is wrong because strict inequalities do not include the endpoint.
Solving $|x|=a$ with only one answer; this is wrong because when $a>0$ , $|x|=a$ gives $x=a$ or $x=-a$ .
Writing $|x|<a$ as $x<-a$ or $x>a$ ; this is wrong because less than means the solution is between the two boundary values, so $-a<x<a$ .
Treating and and or the same way; this is wrong because and means overlap, while or means combine all values from either inequality.

Practice Questions

1 Solve and graph the inequality $3x-7\leq11$ .
2 Solve the compound inequality $-4<2x+6\leq14$ and write the answer in interval notation.
3 Solve the absolute value inequality $|x-3|\geq5$ .
4 Explain why multiplying both sides of $-2x<10$ by $-\frac{1}{2}$ changes the inequality direction.