How to Solve Quadratic Inequalities

How to Solve Quadratic Inequalities

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

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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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