## Thursday, June 9, 2016

### High School Math Solutions – Inequalities Calculator, Quadratic Inequalities

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:
1. Move everything to one side of the inequality sign
2. Set the inequality sign to an equal sign and solve for x
3. Create three intervals
4. Pick a number in each inequality and see if it satisfies the original inequality
5. Select the proper inequality

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

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

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…

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