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Math 105: Precalculus Algebra13 chapters | 103 video lessons

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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.

We also recommend watching Graphing & Solving Quadratic Inequalities: Examples, Lesson & Quiz and Set Notation, Compound Inequalities, and Systems of Inequalities

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.

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 <= 50*t*

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 <= 50*t*

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 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*

Notice the word 'or' in this example. 'Or' means you have a choice; *t* only needs to meet one of the two conditions. It doesn't have to meet both of them. You will only take one of the two trips.

You could solve (simplify) the inequality on the right, so it looks like this:

17 <= *t* or 10 <= *t*

*t* can be either 17 or more months, or it can be 10 or more months. If you are good at imagining numbers, you will see that means that the soonest you can take the trip is after 10 months. By combining the two inequalities, you find that *t* can really be anything 10 or greater. You will be able to choose at least one of the trips if you wait at least 10 months.

Using a number line, you shade the values greater than 17, and then you shade the values greater than 10; however, now, you want the **union** of the two sets. You want all the places on the number line that satisfy one condition, the other condition, or both. So, your solution is any place that has shading of any kind.

Using the number line, you again see that *t* must be 10 or more.

A compound inequality is the combination of two or more inequalities. Sometimes one of the inequalities must be true. In that case, the two inequalities will be combined with the word 'or.' Other times, both inequalities must be true at the same time. In that case, the two inequalities will be combined with the word 'and' (or they will written as a 'between' inequality).

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