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:

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

Looks good!

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

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

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

Step 5: Select the proper inequality

We’ve selected 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