--- title: "Solving Linear Inequalities and Solution Sets" description: "Solve inequalities in one variable with the balancing steps used for equations, flip the sign when multiplying or dividing by a negative, and show the answer on a number line." canonical: https://lightmysky.com/learn/mathematics/solving-linear-inequalities-and-solution-sets-mt_QkG3Cw5VgM source: https://lightmysky.com/learn/mathematics/solving-linear-inequalities-and-solution-sets-mt_QkG3Cw5VgM.md retrieved: 2026-09-02 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Solving Linear Inequalities and Solution Sets Solve inequalities in one variable with the balancing steps used for equations, flip the sign when multiplying or dividing by a negative, and show the answer on a number line. Subject: Mathematics · Area: Algebra · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/solving-linear-inequalities-and-solution-sets-mt_QkG3Cw5VgM ## Ready when they can - Solve 3(x - 2) < 12 and show the solution on a number line - Flip the inequality sign when dividing both sides by a negative number - Solve a double inequality such as -3 < 2x + 1 ≤ 7 and list the integers in it ## Lesson: Solving Linear Inequalities and Solution Sets You have been collecting terms and balancing both sides all through the last few stops, and an inequality is solved with the very same moves. Add, subtract, multiply and divide both sides just like an equation. There is one new rule to keep in your pocket: when you multiply or divide both sides by a negative number, the inequality sign flips. Everything else works exactly as before. *(drawing: The solution x < 4 is a whole stretch of the number line, running left from an open point at 4.)* The answer to an inequality is not a single number but a solution set: every value that makes it true. On a number line, an open circle means the endpoint is not included, which matches the strict signs less than and greater than. A closed circle means the endpoint is in the set, which matches the signs less than or equal to and greater than or equal to. The shaded direction shows which side of the endpoint is the solution. Some answers come in two stretches, like x < -2 or x >= 5: two arms with a gap between them. **Example.** Solve 3(x - 2) < 12. Expand first: 3x - 6 < 12. Add 6 to both sides: 3x < 18. Divide by 3: x < 6. The 3 is positive, so the sign stays. The solution is every number less than 6, which on a number line is a stretch running left from an open circle at 6. Test a value: x = 5 gives 3(5 - 2), which is 9, and 9 is less than 12, so the check passes. **Example.** Now the flip. Solve 8 - 3x <= -4. Subtract 8 from both sides: -3x <= -12. Now divide by -3, which is negative, so the sign flips: x >= 4. The solution is every number from 4 to the right, a closed circle at 4 because the sign is less than or equal to. Test a value: x = 5 gives 8 - 15, which is -7, and -7 is less than or equal to -4, so it works. Test a value outside, like x = 3: 8 - 9 is -1, which is not less than or equal to -4, so 3 is correctly excluded. **Example.** A double inequality pins x between two values at once. Solve -3 < 2x + 1 <= 7. Subtract 1 from all three parts: -4 < 2x <= 6. Divide all three parts by 2: -2 < x <= 3. The solution is the stretch between -2 and 3, open at -2 and closed at 3. If the question asks for the whole numbers in the set, list them: -1, 0, 1, 2 and 3. The strict end leaves -2 out, and the or-equal end keeps 3 in. **Tip.** When in doubt about the direction, test a value. Pick one number from inside your solution set and one from outside, and plug both into the original inequality. The inside one should make it true and the outside one false. It takes ten seconds and catches a flipped sign before the examiner does. **Recap.** Solve an inequality with the same balancing steps as an equation, flip the sign when you multiply or divide by a negative, and show the solution set on a number line with an open or closed endpoint. ## Practice 24 questions on this page, each with its working shown. ## Needs first - [Simultaneous Equations with One Quadratic](https://lightmysky.com/learn/mathematics/simultaneous-equations-with-one-quadratic-mt_D89ql2sJvc) - [Inequalities and their solution sets](https://lightmysky.com/learn/mathematics/inequalities-and-their-solution-sets-mt_i2F1nWxJjv) ## Opens up - [Solving Quadratic Inequalities](https://lightmysky.com/learn/mathematics/solving-quadratic-inequalities-mt_dukJtB_0i8)