Quadratic Inequalities
Solve the equation first, then test intervals.
About this topic
A quadratic inequality such as x² − 3x − 4 > 0 asks where a parabola lies above or below the x-axis. Solve the related equation first to find the boundary points, then test values in each interval, or look at the graph.
For an upward parabola, "less than zero" gives the region between the roots and "greater than zero" gives the outside regions. Here x² − 3x − 4 = (x − 4)(x + 1), with roots −1 and 4, so the solution to the inequality is x less than −1 or x greater than 4. Test a point such as 0: it gives −4, which is negative, confirming the middle region is not part of the answer.
These problems answer questions like for what prices is profit positive, or when is a projectile above a certain height. The answer is usually written in interval notation or on a number line, with open circles for strict inequalities and closed circles when equality is allowed.
Do not simply take the roots as the answer, and do not forget to flip the inequality when multiplying or dividing by a negative. Sketching the parabola is the safest way to see which intervals qualify.
Key ideas
- Roots split the number line
- Test a point in each interval
- Choose the intervals that satisfy the sign
Quick check
Use cases: where Quadratic Inequalities helps
- Homework: review the key idea behind Quadratic Inequalities before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Quadratic Inequalities without notes.
- Visual learning: play with the interactive lab to see how Quadratic Inequalities changes when you adjust its inputs.
- Teaching a friend: explain Quadratic 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 Quadratic Inequalities?
Solve the equation first, then test intervals.
What will I learn in Quadratic Inequalities?
Roots split the number line. Test a point in each interval. Choose the intervals that satisfy the sign.
Is Quadratic 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 Quadratic 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 Quadratic 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.