--- title: "Solving Quadratic Inequalities" description: "Solve inequalities such as x² - x - 6 > 0 by finding the roots, sketching the parabola, and reading off the x values where the curve is on the required side of the axis." canonical: https://lightmysky.com/learn/mathematics/solving-quadratic-inequalities-mt_dukJtB_0i8 source: https://lightmysky.com/learn/mathematics/solving-quadratic-inequalities-mt_dukJtB_0i8.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 Quadratic Inequalities Solve inequalities such as x² - x - 6 > 0 by finding the roots, sketching the parabola, and reading off the x values where the curve is on the required side of the axis. Subject: Mathematics · Area: Algebra · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/solving-quadratic-inequalities-mt_dukJtB_0i8 ## Ready when they can - Find the roots first, then sketch to decide which regions satisfy the inequality - Write the answer as one or two ranges with the correct strict or inclusive signs - Explain why an upward parabola gives two separate ranges for > 0 ## Lesson: Solving Quadratic Inequalities You have already solved linear inequalities: same balancing moves as an equation, flip the sign when you divide by a negative, and the answer is a stretch of the number line with an open or closed endpoint. A quadratic inequality looks the same on the outside, but the balancing moves run out of steam. You cannot just divide by x squared, because x squared can be zero. So the method changes shape: find the roots of the matching equation, sketch the parabola, and read off which stretches of the x axis put the curve on the right side of the axis. *(drawing: A quadratic inequality often answers with two ranges at once, not one stretch.)* The method has four moves, and the roots do all the heavy lifting. First, replace the inequality sign with an equals sign and solve that quadratic equation, by whatever method fits. Those roots are the only x values where the curve can cross the axis, so they are the boundaries of your answer. Second, decide the shape: if the x squared term is positive the parabola opens up, and if it is negative it opens down. If the x squared term is negative, multiply the whole inequality by -1 and flip the sign first, so the curve opens up and the shape rules below just work. Third, check the middle: pick a test value between the two roots, often zero, and see which side of the axis it lands on. Fourth, read off the ranges. For a U shape that is above the axis, the answer is the two stretches outside the roots. For a U shape below the axis, it is the single stretch between them. An n shape flips both of those. The sign of the inequality decides which side you want, and the shape of the curve decides where that side is. **Example.** Solve x squared - 3x - 4 > 0. Set the equals version: x squared - 3x - 4 = 0. The pair that multiplies to -4 and adds to -3 is -4 and 1, so the factors are (x - 4)(x + 1) and the roots are 4 and -1. The curve opens up. Test the middle: x = 0 gives -4, below zero, so the curve sits under the axis between the roots. The inequality wants above the axis, so the answer is the two outer stretches: x < -1 or x > 4. Both signs are strict, so both endpoints stay open. **Example.** Now the other side of the axis. Solve x squared + x - 6 < 0. The equals version factors as (x + 3)(x - 2), so the roots are -3 and 2. The curve opens up, and you want the part that is below the axis, which for a U shape is the middle stretch. The answer is -3 < x < 2. Notice the difference in shape: a below-the-axis U shape gives one range between the roots, and an above-the-axis U shape gives two ranges outside them. Same curve, opposite asks. **Example.** The roots can collapse. Solve x squared + 8x + 16 < 0. The equals version is (x + 4) squared = 0, a repeated root at -4. The curve touches the axis there and never dips below it, so there is no x that makes it strictly less than zero. No solution. If the question had asked x squared + 8x + 16 >= 0 instead, the answer would be every x, because the curve is above the axis everywhere except the touching point. And if an inequality has no roots at all, the curve never crosses the axis, so the whole line is either in or out. **Recap.** Solve the matching equation to find the roots, sketch the parabola in your head, and read off the ranges where the curve is on the required side of the axis, keeping endpoints open or closed to match the sign. ## Practice 24 questions on this page, each with its working shown. ## Needs first - [Solving Linear Inequalities and Solution Sets](https://lightmysky.com/learn/mathematics/solving-linear-inequalities-and-solution-sets-mt_QkG3Cw5VgM) ## Opens up - [Quadratic Sequences and Second Differences](https://lightmysky.com/learn/mathematics/quadratic-sequences-and-second-differences-mt_gyZT3Tcw7U) - [Increasing and Decreasing Functions](https://lightmysky.com/learn/mathematics/increasing-and-decreasing-functions-mt_KOrXxQvBrZ)