Linear Inequalities
Inequalities describe ranges of solutions using <, >, ≤ or ≥.
About this topic
An inequality compares two expressions using signs for less than, greater than, at most or at least, and has a whole range of solutions rather than one. You solve it like an equation, with one critical exception: multiplying or dividing both sides by a negative number reverses the inequality sign.
The solution is shown on a number line with an open circle for strict inequalities and a closed circle when the endpoint is included, or in interval notation such as parentheses for open ends and square brackets for closed ends. Inequalities model limits and budgets, such as the most items you can buy with a fixed amount of money.
Solve −2x + 3 is less than 11. Subtract 3 to get −2x less than 8. Now divide by −2, and because it is negative the sign flips, giving x greater than −4. To check, try x = 0: −2(0) + 3 = 3, which is less than 11, so it works. If you have 50 dollars and tickets cost 8 dollars, then 8n is at most 50 gives n at most 6.25, so you can buy at most 6 tickets.
The flip rule is the most commonly forgotten step. A good sanity check is to pick a test value from your solution set and one from outside it. Another common error is treating the answer like a single number: x greater than 2 means infinitely many numbers, not just 3. Real contexts, like speed limits and minimum scores, use these constantly.
Key ideas
- Solve like equations
- Flip the sign when multiplying or dividing by a negative
- Graph on a number line
Quick check
Use cases: where Linear Inequalities helps
- Homework: review the key idea behind Linear Inequalities before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Linear Inequalities without notes.
- Visual learning: play with the interactive lab to see how Linear Inequalities changes when you adjust its inputs.
- Teaching a friend: explain Linear Inequalities in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Linear Inequalities?
Inequalities describe ranges of solutions using <, >, ≤ or ≥.
What will I learn in Linear Inequalities?
Solve like equations. Flip the sign when multiplying or dividing by a negative. Graph on a number line.
Is Linear Inequalities part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Linear Inequalities lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Linear Inequalities?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.