*Contributor: Mason Smith. Lesson ID: 11243*

You've heard of compound interest but do you have any interest in compound inequalities? These are "and" and "or" inequalities that can be solved with graphs and simple steps. Try these word problems!

categories

subject

Math

learning style

Visual

personality style

Lion

Grade Level

Middle School (6-8), High School (9-12)

Lesson Type

Quick Query

Q: How does a math teacher get a compound fracture?

A: She breaks her hAND.

In this *Inequalities* series, you have tackled a large assortment of inequalities.

Now, it is time to tackle the most difficult of all inequalities: the *compound inequality*!

If you need a review first, catch up in the right-hand sidebar under **Related Lessons**.

For a compound inequality, it is not just one inequality that is being solved, but multiple, and we must be extremely careful with the wording in the problem to ensure that we do not miss important information!

For a compound inequality, we must first split the inequality, then compare them together, accounting for the connecting word *and* or *or*. We will tackle "and" statements first.

For *and *statements, look for the place where the two inequalities overlap; that is the only place where we can have a solution.

For example, consider this compound inequality: x > 5 and x < 7. When we compare the two graphs, x > 5 is:

and x < 7 is:

So, the only place where *both* of the inequalities mathc is from 5 to 7, *non-inclusive*. We can rewrite that as

5 < x < 7, which means *x* must be greater than 5 but smaller than 7.

What if we have two equations that *don't* overlap? For example x > 3 and x < 1. We have *no solution* since a number can't be both larger than 3 and smaller than 1.

Often, we are given one equation with a hidden *and*, for example: 4 < x+2 < 8. This can also be rewritten as

4 < x+2 and x+2 < 8, which are much easier to solve.

We get 2 < x and x < 6, which we can put together again to get 2 < x < 6. So, *x* is between 2 and 6.

A simpler way to solve an "and" statement is to leave it in its original form; for example:

–5 < 2x+3 < 9.

To solve without splitting the inequality apart -- or doing things to both sides like in equations -- we must do it to *all* three parts. So, when we subtract 3, we must subtract 3 from both –5, 2x+3, and 9 to get –8 < 2x < 6. Now, we divide everything by 2 to get –4 < x < 6.

But wait! What if we have to divide or multiply by a negative, as in

–8 < -2x < 6? Instead of flipping the direction of the signs when we divide, we will flip the two outside numbers after you divide to get the right answer!

We end up with –3 < x < 4, which makes sense versus 4 > x.

Now, that we have an idea of how to solve "and" statements, let's move on to "or" statements.

For *or* statements, instead of the correct values being those where both inequalities overlap, as in *and* statements, for *or *statements, the value only has to work for *one* inequality. If *both* work, great, but we only need one.

Let's try –4 + a > 1 or –4 + a < -3.

Just solve for both (a > 5 or a < 1).

There is no way to combine an *or *statement.

Now that we know how to solve for *and* and *or* statements, let's practice solving and graphing them.