How to Solve Quadratic Inequalities? (+FREE Worksheet!)
How to solve quadratic inequalities, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.
Learn how to solve Quadratic Inequalities using similar methods that we use for solving equations.
Solve Quadratic Inequalities: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Solve the boundaryTemporarily treat the inequality like an equation.
- Choose the sideUse the sign or test a number if the direction is not obvious.
- Graph the solutionUse the correct endpoint and shade the values that work.
Worked examples
Flip the sign
- Divide both sides by -3.
- Reverse the inequality sign.
- Simplify 12 divided by -3.
Keep the sign
- Subtract 5 from both sides.
- No negative multiplication or division happened.
- Keep the sign direction.
Try one before moving on
Solve Quadratic Inequalities: pop-up practice
Related Topics
- How to Graph Quadratic Functions
- How to Solve a Quadratic Equation
- How to Graph Quadratic Inequalities
Step-by-step guide to solve Solving Quadratic Inequalities
- A quadratic inequality can be written in one of the following standard forms:
(ax^2+bx+c>0, ax^2+bx+c<0, ax^2+bx+c≥0, ax^2+bx+c≤0) - Solving a quadratic inequality is like solving equations. We need to find solutions.
For education statistics and research
Solving Quadratic Inequalities, Example 1:
Solve quadratic inequality. (x^2-6x+8>0)
Solution:
Factor: (x^2-6x+8>0→(x-2)(x-4)>0)
Then the solution could be (x<2) or (x>4).
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Solving Quadratic Inequalities, Example 2:
Solve quadratic inequality. (x^2-7x+10≥0)
Solution:
Factor: (x^2-7x+10≥0→(x-2)(x-5)≥0). (space) (2) and (5) are the solutions. Now, the solution could be (x≤2) or (x≥5).
Solving Quadratic Inequalities, Example 3:
Solve quadratic inequality. (- x^2-5x+6>0)
Solution:
Factor: (- x^2-5x+6>0→-(x-1)(x+6)>0)
Multiply both sides by (-1: (-(x-1)(x+6))(-1)>0(-1)→(x-1)(x+6)<0) Then the solution could be (-6x) and (x>1). Choose a value between (-1) and (6) and check. Let’s try (0). Then: (- 0^2-5(0)+6>0→6>0). This is true! So, the answer is: (-6<x<1)
Solving Quadratic Inequalities, Example 4:
Solve quadratic inequality. (x^2-3x-10≥0)
Solution:
Factor: (x^2-3x-10≥0→(x+2)(x-5)≥0. -2) and (5) are the solutions. Now, the solution could be (x≤-2) or (x≥5).
Exercises for Solving Quadratic Inequalities
Solve each quadratic inequality.
- (color{blue}{x^2+7x+10<0})
- (color{blue}{ x^2+9x+20>0})
- (color{blue}{x^2-8x+16>0})
- (color{blue}{ x^2-8x+12≤0})
- (color{blue}{ x^2-11x+30≤0})
- (color{blue}{ x^2-12x+27≥0})
Download Solving Quadratic Inequalities Worksheet
- (color{blue}{-5<x<-2})
- (color{blue}{x<-5 or x>-4})
- (color{blue}{x<4 or x>4})
- (color{blue}{2≤x≤6})
- (color{blue}{5≤x≤6})
- (color{blue}{x≤3 or x≥9})
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