How to Solve Quadratic Inequalities

How to Solve Quadratic Inequalities

Learn how to solve Quadratic Inequalities using similar method that we use for solving equations.

Step by step guide to solve Solving Quadratic Inequalities

  • A quadratic inequality is one that 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.

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 \(x>4\).

Solving Quadratic Inequalities – Example 2:

Solve quadratic inequality. \(x^2-7x+10≥0\)

Solution:

Factor: \(x^2-7x+10≥0→(x-2)(x-5)≥0\). \(2\) and \(5\) are the solutions. Now, the solution could be \(x<2\) or \(x=2\) and \(x=5\) 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 \(-2≤x≤5\) or \(-6≥x\) and \(x≥1\). Let’s choose zero to check:
\(0^2-3(0)-10≥0→-10≥0\), which is not true. So, \(-6≥x\) and \(x≥1\)

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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