How To Solve Quadratic Inequalities?

 

By Robert O

This tutorial assumes that you are through with lessons on factoring quadratic equations and solving quadratic equations. A quadratic inequality is just like a quadratic equation where the inequality sign replaces the equal sign. As such, the solution of quadratic inequalities follows the same method for solving any quadratic equation.

The two roots from a quadratic equation are the solution of a quadratic inequality. The only thing you will have to do is test the roots if they are indeed the solutions to the inequality. As a general rule, always toggle the inequality sign when multiplying or dividing through by a negative quantity.

A quadratic inequality takes the form ax² + bx + c < 0. The inequality sign can be any of the four that we already know.

What are the steps to solve quadratic inequalities?

  • First, change the inequality sign to an equal sign.
  • Secondly, find the roots using an appropriate formula for solving a quadratic equation.
  • Lastly, find which of the roots satisfy the inequality.

Example 1:

Solve for x in the equation: x² – 5 x + 6 ≥ 0

Solution

The equation is already in the form of ax² + bx + c = 0.  The next task is to substitute the inequality by an equal sign then solve for x.

    \[x^{2}-5 x+6=0\]

Using the quadratic formula

    \[x=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a}\]

  the two roots are:

    \[x=\frac{+5 \pm \sqrt{25-24}}{2}=3 \text { or } 2\]

2 and 3 are solutions to the problem and the boundary limits. Any value that is less than 2 or greater than three are also solutions. Take any value and test.

The graph below shows a complete solution:

Example 2:

Solve for x in the equation: 2 x² + x – 15 ≤ 0

Solution

Rewriting it in the quadratic form:

    \[2 x^{2}+x-15=0\]

Now use the quadratic formula or any other that you are familiar with to find out that the roots are -3 and +2.5.

The values of x are -3 or +2.5.

Testing with any value less than 3, say 4, does not satisfy the inequality. A value greater than 2.5, say 3, also does not satisfy the inequality. Any values between -3 and +2.5 satisfy the inequality and are the solutions to our problem.

Graphically:

 

Example 3:

Solve for x in the equation: 2 x² – 7 x + 3 > 0

Solution

The roots of the quadratic equations are 0.5 and 3. These are the boundaries. So, let’s test them if they satisfy the inequality.

Both 0.5 and 3 are not part of the solution. They only act as the boundary. So, what are the solutions to the inequalities? Let’s test with a value greater than 3 and less than 0.5. We also test with a value between 3 and 0.5.

All values from negative infinity to 0.5 and values from 3 to positive infinity satisfy the inequality. Since the inequality sign was strictly greater than, we will not shade the dots this time.

Graphically:

 

Remarks

Solving quadratic inequalities is just as easy as solving quadratic equations. The only trick is that you need to find sets of values that satisfy the inequality after getting the roots.

About the Author

This lesson was prepared by Robert O. He holds a Bachelor of Engineering (B.Eng.) degree in Electrical and electronics engineering. He is a career teacher and headed the department of languages and assumed various leadership roles. He writes for Full Potential Learning Academy.