# Compound Inequality: Definition & Concept

Instructor: Kimberlee Davison

Kim has a Ph.D. in Education and has taught math courses at four colleges, in addition to teaching math to K-12 students in a variety of settings.

A compound inequality is simply the combination of two or more inequalities. In this lesson, you will learn about different ways two inequalities can be combined and how to find their solution using a number line.

## What is a Compound Inequality?

A compound inequality is a combination of two or more inequalities joined by either an 'and' or an 'or.' In mathematics, most often we deal with equalities - two statements that must, at all times, remain exactly equal to each other; however, sometimes we are interested in more than or less than relationships. We have two statements, and one must always be either bigger than or at least as big as the other.

## Let's Take a Trip

Imagine, for example, that you are planning a vacation to Memphis. Maybe you have \$500 set aside for the trip. Or, maybe you are saving a little towards it monthly - the amount you will have to spend will depend on how long you have until you go. In either case, you must spend on the trip no more than the amount you have saved. Going into debt isn't an option. You don't have to spend your entire savings, but you can't spend more than that.

Mathematically, you have:

Trip budget <= Savings

(The '<=' here means 'less than or equal to.')

You may end up with a complex expression on each side of the '<=' symbol (the inequality). For example, maybe you use the variable t to represent time. Each month you save \$50 towards your trip. The inequality now becomes:

Trip budget <= 50t

So, if you save for 6 months, then t = 6. Your savings would be \$300, and your trip budget would have to be less than or equal to \$300.

Now pretend that you have visited a travel agent and have learned that the cost of your trip is going to be \$850. You want to know how many months you will need to save before you have enough money to make the trip. In other words, you need to solve for t in the following inequality:

850 <= 50t

In most cases, you can solve for t exactly as if you had an equation instead of an inequality. Simply divide both sides by 50. You get: 17 <= t. Generally, we reverse the order, as we like to see the variable on the left side: t >= 17. In other words, you need to save for the trip for at least 17 months.

## Compound Inequalities

Compound inequalities are needed when you have two conditions to be met, instead of just one. For example, consider the following possible additions to our story.

A. The trip must happen less than 2 years from now (24 months).

Here, your two inequalities (conditions) are:

17 <= t and t < 24

The word 'and' between the two conditions is very important. It means that you don't get to choose between the two conditions. Both conditions must be met. Another way to say this is that we need the intersection of the two solution sets. We need the values of t that work for both inequalities.

In some cases, it is easy to picture the solution. t must be 17 or larger. It must also be less than 24. Therefore it can be any value from 17 to just under 24 months.

Other times, it may be trickier to imagine the solution. In that case, a number line, as shown here, might be helpful. You shade the values that work for the first inequality, as well as those that work for the second. The place they overlap is the set of numbers that works for both inequalities.

In this case, you have a between situation. The solution is all values between two numbers. When that happens, you can combine the two inequalities, so that they look like this:

17 <= t < 24

B. There are a lot of different trip situations that also would use more than one inequality. Let's say that you have the choice to either save \$50 per month for the \$850 trip or to save \$100 per month for a \$1000 trip.

Your two inequalities now look like this:

17 <= t or 1000 <= 100 t

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