We’ve learned how to solve

linear inequalities. Now, it’s time to learn how to solve quadratic inequalities. Solving quadratic inequalities is a little harder than solving linear inequalities. Let’s see how to solve them.

There are a couple ways to solve quadratic inequalities depending on the inequality. I’ll focus on explaining the more complicated version.

We’re given the quadratic inequality:

x^2+2x-8\le0
Here are the steps to solving it:

- Move everything to one side of the inequality sign
- Set the inequality sign to an equal sign and solve for x
- Create three intervals
- Pick a number in each inequality and see if it satisfies the original inequality
- Select the proper inequality

Now we will go through this example step by step to understand a little better how to solve it.

Step 1: Make sure you start with 0 on one side

x^2+2x-8\le0

Looks good!

Step 2: Set the inequality sign to an equal sign and solve for x

x^2+2x-8=0

(x-2)(x+4)=0

x=2\:and\:x=-4

Step 3: Create three intervals

We are able to pick three intervals from looking at the number and seeing where the function crosses the x-axis (i.e. where the function is equal to 0).

Step 4: Pick a number in each inequality and see if it satisfies the original inequality

x<-4 -4<x<2 x>2

Table Header |
x<-4 |
-4<x<2 |
x>2 |

x^2+2x-8 |
(-5)^2+2(-5)-8=7 |
(0)^2+2(0)-8=-8 |
(3)^2+2(3)-8=7 |

Sign |
\quad\quad\quad\quad+ |
\quad\quad\quad\quad- |
\quad\quad\quad\quad+ |

We’ve turned the intervals into inequalities. Then, we picked a number in each inequality to see if it satisfied the original inequality,

x^2+2x-8\le0.

Step 5: Select the proper inequality

-4<x<2

-4\lex\le2

We’ve selected

-4<x<2 because it satisfies the original inequality because the quadratic is negative when x is between -4 and 2. However, don’t forget that the original equality contains the ≤ symbol so that means x can equal 0 too. We change the inequality signs because we know -4 and 2 are the zeros of the quadratic.

Let’s see some more examples…

Second example (

click here):

Last example… This one is simple (

click here):

Hopefully, that wasn’t too hard! Solving quadratic inequalities sometimes require patience to write everything out. For more help check out Symbolab’s practice.

Until next time,

Leah