Solving Linear Inequalities
Lessons in this chapter
About Solving Linear Inequalities
Solving Linear Inequalities takes the one-step and multi-step inequality skills from middle school and extends them to compound and absolute value inequalities.
Writing and Graphing Inequalities and Solving One-Step Inequalities and Solving Multi-Step Inequalities review the core inequality-solving process, including the rule that the inequality sign flips when multiplying or dividing by a negative number.
Solving Compound Inequalities covers two inequalities joined by 'and' or 'or': an 'and' compound inequality needs both conditions true at once (a value between two numbers), while an 'or' compound inequality needs at least one condition true (a value outside a range).
Solving Absolute Value Inequalities combines both ideas: an inequality like |x| < 5 becomes the compound inequality -5 < x < 5, while |x| > 5 splits into x < -5 or x > 5, since absolute value inequalities always resolve into one of these two compound forms.
Each of the five lessons here is an Algebra 1 worksheet worked in the page, so Writing and Graphing Inequalities, Solving Compound Inequalities and Solving Absolute Value Inequalities each carry their own set of math problems. Answers are checked as you enter them, which matters most on the questions where multiplying by a negative flips the inequality sign and the solution set quietly reverses. The practice is free and nothing is locked behind an account.
Frequently asked questions
Writing and Graphing Inequalities starts with reading an inequality and showing its solution on a number line. Solving One-Step Inequalities and Solving Multi-Step Inequalities then use the same inverse operations as equations, with one new rule: multiplying or dividing by a negative reverses the sign. Solving Compound Inequalities handles two conditions joined by and or or, and Solving Absolute Value Inequalities reuses the two-case split from the previous chapter, this time producing a range of solutions rather than two exact values. The chapter sits in the high school Algebra conceptual category (A).
A compound inequality joins two inequalities with "and" (both must be true, giving a range between two values) or "or" (at least one must be true, giving two separate ranges).
Solve both inequalities separately, then find the overlap between their solution sets: the values that satisfy both conditions at once, often written as a single range like 2 < x < 7.
Solve both inequalities separately, then combine both solution sets: any value that satisfies at least one of the two conditions is part of the answer.
|x| < 5 becomes the "and" compound inequality -5 < x < 5, while |x| > 5 becomes the "or" compound inequality x < -5 or x > 5: the direction of the original inequality determines which type of compound inequality results.
Multiplying or dividing by a negative number reverses the order of every number on the number line, so flipping the sign is what keeps the statement true after that reversal.