High School Math Solutions – Inequalities Calculator, Compound Inequalities

In the previous post, we talked about solving linear inequalities. In today’s post we will focus on compound inequalities, which are just a little more complicated than linear inequalities.

A compound inequality is an equation of two or more inequalities joined together by “and” or “or”. Sometimes you see compound inequalities joined by “and” written like -3<2x-1<5. Compound inequalities have three parts: the left, the middle, and then right. Just like compound inequalities, our goal is to isolate the variable. However, we want the variable to be isolated in the middle part.

-3<2x-1<5\quad\quad\quad\quad\to\quad\quad\quad-1<x<3

How do we get that answer? The key is to break the compound inequalities.

Here are the steps for solving compound inequalities:

1. Break the inequality into two parts
2. Solve each linear inequality by isolating the variable
3. Combine the inequalities 

Simple enough? Let’s break down an example step by step to understand the concept.

Here’s our problem (click here):

-17<3+10x\le33

1. Break the inequality into two parts

            -17<3+10x                                                    3+10x\le33

Based on the definition of a compound inequality, we know that this compound is made of two inequalities. We break the compound inequality down into its two inequalities.

2. Solve each linear inequality by isolating the variable

              -17<3+10x                                                    3+10x\le33
                -20<10x                                                        10x\le30

                 -2<x                                                              x\le3

Using algebra, we can isolate x and get two simplified inequalities.


3. Combine the inequalities
                 -2<x                                                           x\le3
                                                 -2<x\le3

Make sure after combining the inequalities that the simplified compound inequality makes logical sense.


Let’s see two more examples . . .

Second example (click here)

Last example (click here)

Solving compound inequalities is just little harder than solving linear inequalities. Remember to keep your work nice and neat and practice and you will do great! For more practice on compound inequalities, check out Symbolab’s practice.

Until next time,

Leah