How to Solve Quadratic Inequalities? (+FREE Worksheet!)
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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