Writing and Using Inequalities

Learning Objective

• Develop inequalities to represent real world situations and use them to solve problems.

InequalitiesA math sentence that defines a range of numbers; inequalities contain the symbols `<`, `<=`, `>`, or `>=`. are used all the time in the world around us—we just have to know where to look. Figuring out how to interpret the language of inequalities is an important step toward learning how to solve them in everyday contexts.

You are confronted with mathematical inequalities almost every day, but you may not notice them because they are so familiar. Think about the following situations: speed limits on the highway, minimum payments on credit card bills, number of text messages you can send each month from your cell phone, and the amount of time it will take to get from home to school. All of these can be represented as mathematical inequalities. And, in fact, you use mathematical thinking as you consider these situations on a day-to-day basis.

When we talk about these situations, we often refer to limits, such as “the speed limit is `65` miles per hour” or “I have a limit of `250` text messages per month.” However, we don’t have to travel at exactly `65` miles per hour on the highway, or send and receive precisely `250` test messages per month—the limit only establishes a boundary for what is allowable. Thinking about these situations as inequalities provides a fuller picture of what is possible.

Consider this problem:

An `18`-wheel truck stops at a weigh station before passing over a bridge. The weight limit on the bridge is `65,000` pounds. The cab (front) of the truck weighs `20,000` pounds, and the trailer (back) of the truck weighs `12,000` pounds when empty. In pounds, how much cargo can the truck carry and still be allowed to cross the bridge?

This problem offers us an upper limit—`65,000` pounds—but we are interested in finding out the full range of possibilities for the weight of the truck’s cargo. We can represent the situation using the following inequality, where `c` is the weight (in pounds) of the truck’s cargo:

Solving this inequality for `c`, we find that `c<=33,000`. This means that the weight of the cargo in the truck can be anywhere between `0` pounds and `33,000` pounds and the truck will be allowed to cross the bridge.

When you are solving or building these inequalities, it is important to know which inequality symbol you should use. Watch for certain phrases that will tip you off:

Many problems, though, will not explicitly use words like “at least” or “is less than.” So how do you figure out which symbol is appropriate in a given situation?

The key is to think about the context of the problem, and to relate the context to one of the situations listed in the table. Context refers to the real-life situation in which the problem takes place.

So, for example, think back to the truck problem that we solved above. The maximum weight allowed on this bridge was `65,000` pounds. We can also think of this relationship using the language of inequalities used in the table: the total weight of the cab, trailer, and cargo had to be no more than `65,000` pounds. Once we have identified the relationship between the two quantities we can put in the appropriate symbol.

Consider another problem that requires us to look at context:

Erykah has found three pairs of running sneakers that she likes that cost `$150`, `$159`, and `$179`. She has saved `$31` already, and she has a job where she earns `$8.50` per hour. How many hours will she have to work in order to afford any of these sneakers?

Notice that this problem is not asking us to find out the number of hours Erykah will have to work to afford a specific sneaker—it is asking about any of these sneakers. Since the least expensive sneaker is `$150`, we need to create an inequality that shows how much she needs to work in order to have at least `$150`—or, in math symbols, `>=$150`. We will use the variable `h` to represent the number of hours she works.

We can solve this inequality like this:

Erykah needs to work at least `14` hours to afford any of the running sneakers. If she works more hours, she’ll be able to afford the more expensive models.

Some real-life inequalities require the use of absolute value. Once again, there are certain phrases to look for in a description of these kinds of situations:

As before, you’ll still need to consider the context of the problem to be sure if absolute value is appropriate. Here’s an example:

At a tannery, a worker uses a machine to cut large strips of leather down to a consistent size in order to make belts. The belts are supposed to be `35` inches long, but the worker is allowed a margin of error that is no more than `1/10` inch for the belt to be of acceptable length. Write an inequality that represents the range of acceptable belt lengths.

We know that the belts have to be `35` inches long, give or take up to `1/10` inch. How do we account for this margin of error? We use an absolute valueThe value of a number without regard to its sign. inequality, like this:

In this inequality, the term ` |35 - b|` represents the absolute value of the difference between the target length of the belt and the actual length of the belt. It does not matter if the belt is a little longer or a little shorter—either is acceptable. Note that the order of these terms could have been reversed, and we could have written ` |b - 35|`. We care about distance, but not direction. We used the inequality symbol `<=`, because the difference must be less than or equal to the margin of error.

Let’s solve the inequality:

Any value between and including `34 9/10` and `35 1/10` will satisfy this inequality—this represents the acceptable range of values for the belt lengths.

Inequalities can be used to model a number of real-life situations. When converting such word problems into inequalities, begin by identifying how the quantities relate to each other, and then pick the inequality symbol that is appropriate for that situation. When solving these problems, remember that the solution will be a range of possibilities—inequalities do not have a single answer, as equations do. Absolute value inequalities can be used to model situations where margin of error is a concern.