Writing, Solving, and Graphing Inequalities in One Variable

Learning Objectives

• Solve algebraic inequalities in one variable using a combination of the properties of inequality.
• Represent inequalities on a number line.

Sometimes finding a range of possible values for a situation is more appropriate than finding a single value. When you are driving on a highway and see a “Speed Limit `65`” sign, you know that it doesn’t mean you must drive exactly `65` mph. You could go `64`. Driving `59.5` would work. A speed of `55` might get you some honks and angry gestures, but not a traffic ticket. There’s a whole bunch of legal speeds you could drive, not just one. In cases like this with more than one right answer, we use inequalitiesA math sentence that defines a range of numbers; inequalities contain the symbols `<`, `<=`, `>`, or `>=`. , not equations, to represent the situation.

Inequalities are math statements that define a range of values. They are easily recognizable because they contain the symbols `<`, `<=`, `>`, or `>=`.

Inequalities are different than equations, although you can apply what you know about equations to help you understand inequalities. Inequalities and equations are both math statements that compare two values.

An equation contains the symbol `=`, which links two expressions that have the same value. You are familiar with equations like these:

`26 = 21 + 5`

`y=3x+b` 

`5t = 2(t + 3)`

Even without solving them, you know that the quantity on the left-hand side of the equals sign has the same value as the quantity on the right-hand side of the equals sign.

Inequalities are different. In an inequality, one side of the inequality can be larger or smaller than the quantity on the other side. The math symbols `<`, `<=`, `>`, and `>=` provide information about the relative sizes of the two expressions.

The important thing about inequalities is that there are multiple possible solutions. For example, the inequality “`50>=` the number of stars on any United States flag” is a true statement for every American flag ever flown—no flag has had over `50` stars. It holds true for the flag as it was designed in `1777` (`13` stars, `50>=13`), as it looked in `1850` (`30` stars, `50>=30`), and how we see it today (`50` stars, `50>=50`).

Note that the inequality `x>y` can also be written as `y<x`. The sides of any inequality can be flipped as long as the inequality symbol between them is also reversed.

One way to represent inequalities is by using a number line. In the examples below, the range of true values for the inequality is shown in red. An open dot is used to represent `<` and `>` relationships; this symbol indicates that the point on the number line is not included within the range of possible values for the inequality. A closed dot is used to represent `<=` and `>=`, when the two sides of the inequality could be equal.

Stan has more than `$3.50` in his pocket.

The temperature is greater than `-4^@` and less than `12^@`.

`t<=19` 

Just as with equations, there are a number of Properties of InequalityA set of rules for inequalities that describe how addition, subtraction, multiplication, or division can be applied to both sides of an inequality in order to produce an equivalent inequality. that help us work with these types of relationships.

Let’s start with addition and subtraction and the simple inequality `a>b`. If we want to add a quantity `c` to the left side, we also have to add it to the right side in order to keep the inequality true. We can write this property as:

If `a>b`, then `a + c>b+c`. 

People’s ages serve as a good real-life example to model this property. For instance, imagine that you know two people: Adam and Bernard. You know that Adam is older than Bernard (although you do not know how much older). In a certain number of years from now, will Adam still be older than Bernard? Of course! Adam is older to begin with, and they are aging the same amount. In algebraic fashion, you could represent this inequality as:

If Adam's age `>` Bernard's age,

then Adam's age `+` some years `>` Bernard's age `+` the same number of years.

The Subtraction Property is similar. If we begin with the inequality `a>b` again and we subtract `c` from `a`, then we also need to subtract `c` from `b` in order to maintain the relationship. We can write this property as:

If `a>b`, then `a-c>b-c`.

The age example can help you make sense of this relationship as well: if Adam is older than Bernard now, then five years ago Adam was also older than Bernard (because Bernard was also five years younger). You could represent this inequality as:

If Adam's age `>` Bernard's age,

then Adam's age `-` `5` years `>` Bernard's age `-` `5` years.

Not surprisingly, there are Multiplication and Division Properties of Inequality, as well. Let’s look at our standard `a>b` relationship to see how these work out.

A number line can help model what is going on when `c>0`, as well as why the inequality sign “flips” when `c<0`.

Let’s plot two integers on a number line, `5` and `2`. We know that `5>2`. What happens if we multiply both numbers by the same value `c`? The exercise below will let us find out. On this number line, points `a` and `b` are our original values of `2` and `5`. `a>b`.

Now let’s move the slider bar marked `c` to see what happens when we multiply both numbers by the same amount.

If we make `c = 0.5`, we see that `2.5>1`, and `ac>bc`. The inequality has been maintained. While this is a simple example, it holds true for all values where `c>0`. This is written formally as:

If `a>b`, then `ac>bc`, if `c>0`.

If we take the same two numbers and multiply them by `-0.5`, something different happens. The resulting value of `ac` (`-2.5`), is further to the left than the value of `bc` (`-1`). The inequality has been reversed, and `ac<bc`. This pattern holds true for all inequalities—if they are multiplied by a negative number, the inequality flips. This is written formally as:

If `a>b`, then `ac<bc`, if `c<0`. 

The Division Properties of Inequality work the same way. If we divide both sides by a positive number, the inequality is preserved. To put it mathematically:

If `a>b`, then `a/c>b/c`, if `c>0`. 

If we divide both sides of an inequality by a negative number, the inequality is reversed.

If `a>b`, then `a/c<b/c`, if `c<0`. 

The Properties of Inequality will come in handy when you want to solve problems. For example, here is a problem where we can use the Subtraction Property to help us find a range of possible solutions:

In `7` years, Ellie will be old enough to vote in an election. (You must be at least `18` years old to vote.) What can you say about how old she is now?

We could write this inequality as: `e + 7>=18`, where `e` represents Ellie’s age. We can then use the Subtraction Property of Inequality to solve for `e`.

So Ellie’s current age must be at least `11` if she will be able to vote in `7` years. Notice that we are not saying that Ellie is `11` years old now—she could be `11`, or she could be `32`, or she could be `55`. She could be any age as long as that age is `11` or older.

Let’s try another:

In the inequality `-4y>30`, solve for `y`.

To solve this problem, we want to isolate the variable `y`, just as we would to solve an equation. We can do this by dividing both sides of the inequality by `-4`. From the Division Property of Inequality, we know that if we divide both sides of an inequality by a number less than `0`, the inequality will “flip,” so we need to be careful to change the symbol.

We have our answer: `y<-7.5`. But we should check at least one value to make sure that our answer is correct. Let’s pick a value for `y` that is less than `-7.5` and substitute it into the original inequality to see if it works. We’ll use `-10`: 

Since `40` is indeed greater than `30`, `-4y>30` is true for `y = -10`. Based on our test, we can feel confident that our answer is correct. Any value less than `-7.5` will satisfy this inequality.

As we’ve seen, we can solve simple inequalities without much difficulty as long as we use the Properties of Inequality to help us determine the direction of the inequality symbol.

Solving inequalities, then, is very similar to solving equations—what we do to one side we need to do to the other side in order to maintain the “balance” of the inequality. The Properties of Inequality can help us understand how to add, subtract, multiply, or divide within an inequality.

As with one-step inequalities, we can solve two-step inequalities by manipulating the inequality so that we isolate the variable. Observe how to solve a two-step problem:

As with one-step inequalities, we should always substitute values into the original inequality to check the answer. In this problem we found that `k<1/2`, so we should pick a value that is less than `1/2`. Let’s choose `0`. Go ahead and plug that into the original equation and see if it works.

It worked, didn’t it? We got it right the first time.

Inequalities model problems that have a range of answers. They can be mapped along a number line, and they can be manipulated to simplify or solve them. When solving inequalities, it is important to follow the Properties of Inequality:

Addition and Subtraction Properties of Inequality

If `a>b`, then `a+c>b+c` 

If `a>b`, then `a-c>b-c` 

Multiplication and Division Properties of Inequality

If `a>b`, then `ac>bc`, if `c>0` 

If `a>b`, then `ac<bc`, if `c<0` 

If `a>b`, then `a/c>b/c`, if `c>0` 

If `a>b`, then `a/c<b/c`, if `c<0` 

These properties help us manage the inequality symbol properly. The inequality is maintained by addition and subtraction, and by multiplication and division with positive numbers. The inequality is flipped by multiplication and division by negative numbers.