Solving and Graphing Absolute Value Inequalities

 

Learning Objective

·         Solve absolute value inequalities in one variable using the Properties of Inequality.

·         Represent absolute value inequalities on a number line.

 

Introduction

 

The absolute value of a value or expression describes its distance from 0, but it strips out information on the sign of the number or the direction of the distance. Absolute value is always positive or zero, and a positive absolute value could result from either a positive or a negative original value.

 

When solving and graphing absolute value inequalities, we have to consider both the behavior of absolute value and the Properties of Inequality. These types of inequalities behave in interesting ways—let’s get started.

 

Absolute Value: A Primer

 

We know that the absolute value of a number is a measure of size but not direction. So, for example, |27| and |-27| are both 27—absolute value indicates the distance from 0, but doesn’t bother with the direction.

 

Watching a weather report on the news, we may hear “Today’s high was 72°, but we’ll have a 10° swing in the temperature tomorrow. I’ll let you know which way we’re going after these commercials.” Based on this information, tomorrow’s high could be either 62° or 82°. The weatherman has said the difference between the temperatures, but he has not revealed in which direction the weather will go.

 

Finding the absolute value of signed numbers is pretty straightforward—just drop any negative sign.

 

Consider |m| = 7.5, for instance. We know the absolute value of m, but the original value could be either positive or negative. So in this case we say that m = 7.5 or -7.5. The graph below shows |m| = 7.5 mapped on the number line. Notice that we’ve plotted both possible solutions.

 

 

 

Travis is 14 years old. There is a 5 year difference between Travis’ age and his sister’s age, and a 2 year difference between Travis’ age and his brother’s age. Which set of numbers represents all of the possible ages of Travis and his siblings?

 

A) 9, 12, 14, 16, 19

B) 12, 14, 19

C) 9, 14, 16

D) 14, 16, 18, 19, 24

 

Show/Hide Answer

A) 9, 12, 14, 16, 19

Correct. Travis is 14, and his sister is either 5 years older or 5 years younger than him, so she could be 9 or 19. There is a 2 year difference between Travis and his brother, so he could be either 12 or 16.

 

B) 12, 14, 19

Incorrect. Travis is 14, and while his sister could be 19, she could also be 9. Similarly, his brother could be 12, or he could be 16—we don’t know whether his siblings are older or younger, so we have to include all possibilities. The correct age range is 9, 12, 14, 16, 19.

 

C) 9, 14, 16

Incorrect. Travis is 14, and while his sister could be 9, she could also be 19. Similarly, his brother could be 16, or he could be 12—we don’t know whether his siblings are older or younger, so we have to include all possibilities. The correct age range is 9, 12, 14, 16, 19.

 

D) 14, 16, 18, 19, 24

Incorrect. We know that Travis is 14, and his sister is either 5 years older or 5 years younger—so she could be 9 or 19. Likewise, his brother is either 2 years older or 2 years younger, so he could be either 12 or 16. The correct age range is 9, 12, 14, 16, 19.

 

 

 

Absolute Value Inequalities on the Number Line

 

Once the equal sign is replaced by an inequality, graphing absolute values changes a bit. Let’s stick with the example from above, |m| = 7.5, but change the sign from = to ≤. Now we have an absolute value inequality: |m| ≤ 7.5.

 

Think about this weather report: “Today at noon it was only 0°, and the temperature changed at most 7.5° since then.” Notice this does not say which way the temperature moved, and it does not say exactly how much it changed—it just says that, at most, the temperature has changed 7.5°. We can represent this idea with the statement |change in temperature| ≤ 7.5°.

 

To find out the full range of m values that satisfy this inequality, we need to evaluate both possibilities for |m|: m could be positive or m could be negative. If m is positive, then |m| and m are the same number.  If m is negative, then |m| is the opposite of m, that is, |m| is -m. So in this case we have two possibilities, m ≤ 7.5 and -m ≤ 7.5. We need to solve for both:

 

Example

Problem  |m| ≤ 7.5

Answer:   m

7.5

-m

7.5

 

 

 

-m • -1

7.5 • -1

 

 

 

Answer:       m

-7.5

 

It’s important to remember something here: when you multiply both sides of an inequality by a negative number, like we just did to turn -m into m, the inequality sign flips. If you forget to do that, you’ll be in trouble.

 

So m could be less than or equal to 7.5, or greater than or equal to -7.5. Notice that the range of solutions includes both points (-7.5 and 7.5) as well as all points in between. Here is a graph of the inequality on a number line:

 

 

 

We could say “m is greater than or equal to -7.5 and less than or equal to 7.5.” If m is any point between -7.5 and 7.5 inclusive on the number line, then the inequality |m| ≤ 7.5 will be true. We can write this as -7.5 ≤ m ≤ 7.5. This notation places the value of m between those two numbers, just as it is on the number line.

 

Let’s look at a different sort of situation. We’ll evaluate the absolute value inequality |g| > 4.

 

Example

Problem   |g| > 4

Answer:

g

> 

4

-g

> 

4

 

 

 

-g • -1

< 

4 • -1

 

 

 

Answer:

g

-4

 

We find that g could be greater than 4 or less than -4. If we map both those possibilities on a number line, it looks like this:

 

 

The graph shows one ray (a half-line beginning at one point and continuing to infinity) beginning at -4 and going to negative infinity, and another ray beginning at +4 and going to infinity. We could say “g is less than -4 or greater than 4.” That can be written algebraically as -4 >g > 4. This notation tells us that the value of g could be anything except what is between those numbers.

 

Notice the difference between this graph and the graph of |m| ≤ 7.5. In |m| ≤ 7.5, the range of possibilities that satisfied the inequality lies between the two points. In |g| > 4, however, the range of possible solutions lies outside the points, and extends to infinity in both directions.

 

A quick way to identify whether the absolute value inequality will be graphed as a segment between two points or as two rays going in opposite directions is to look at the direction of the inequality sign in relation to the variable.

 

 

Identifying the graphs of absolute value inequalities

 

If the absolute value of the variable is less than the constant term, then the resulting graph will be a segment between two points.

 

If the absolute value of the variable is more than the constant term, then the resulting graph will be two rays heading to infinity in opposite directions.

 

 

For example, think about the inequality |x| ≤ 2, which could be modeled by someone walking a dog on a two-foot long leash. The dog can pull ahead up to the entire length of the leash, or lag behind until the leash tugs him along. He cannot be farther away from the person than two feet in either direction. In other words, the dog can only be at a distance less than or equal to the length of the leash. In the language of algebra, the location of the dog can be described by the inequality -2 ≤ x ≤ 2. The constant is the maximum value, and the graph of this will be a segment between two points.

 

Now consider the opposite inequality, |x| ≥ 2. Imagine a high school senior who wants to go to college two hours or more away from home. He may choose a school three hours east, or five hours west—he’ll go anywhere, as long as it is at least 2 hours away. There is no upper limit to how far he will go. In mathematical terms, the situation can be written as the inequality -2 ≥ x ≥ 2. The constant is the minimum value, and the graph of this situation will be two rays that head out to negative and positive infinity and exclude every value within 2 of the origin.

 

Camille is trying to find a solution for the inequality |d| ≤ 0.5. What can she expect the graph of this inequality to look like?

 

A) A ray, beginning at the point 0.5, going towards negative infinity.

B) Two rays: one beginning at 0.5 and going towards positive infinity, and one beginning at -0.5 and going towards negative infinity.

C) A ray, beginning at the point 0.5, going towards positive infinity.

D) A segment, beginning at the point 0.5, and ending at the point -0.5.

 

Show/Hide Answer

A) A ray, beginning at the point 0.5, going towards negative infinity.

Incorrect. A ray beginning at the point 0.5 and going towards negative infinity is the inequality d ≤ 0.5. This question concerns absolute value, so you must also consider the possibility that -d ≤ 0.5. The correct graph is a segment, beginning at the point 0.5, and ending at the point -0.5.

 

B) Two rays: one beginning at 0.5 and going towards positive infinity, and one beginning at -0.5 and going towards negative infinity.

Incorrect. This number line represents |d| ≥ 0.5. Since the inequality actually had the absolute value of the variable as less than the constant term, the right graph will be a segment between two points, not two rays. The correct graph is a segment, beginning at the point 0.5, and ending at the point -0.5.

 

C) A ray, beginning at the point 0.5, going towards positive infinity.

Incorrect. A ray beginning at the point 0.5 and going towards positive infinity describes the inequality d ≥ 0.5. This question concerns absolute value, so the number line must show that -0.5 ≤ d ≤ 0.5. The correct graph is a segment, beginning at the point 0.5, and ending at the point -0.5.

 

D) A segment, beginning at the point 0.5, and ending at the point -0.5.

Correct. The range of possible values for d includes any number that is less than 0.5 and greater than -0.5, so the graph of this solution set is a segment between those two points.

 

 

Solving One- and Two-Step Absolute Value Inequalities

 

The same Properties of Inequality apply when solving an absolute value inequality as when solving a regular inequality. The main difference is that in an absolute value inequality, you need to evaluate the inequality twice to account for both the positive and negative possibilities for the variable.

 

Let’s start with a one-step example: 3|h| < 21. The first step is to isolate the absolute value term on one side of the inequality. We can do that by dividing both sides by 3, just as we would do in a regular inequality.

 

Example

Problem

3|h|

21

 

|h|

7

 

 

Answer:

h < 7

-h

< 

7

 

-h-1

> 

7 • -1

 

Answer:

h

> 

-7

 

With the inequality in a simpler form, we can evaluate the absolute value as h < 7 and h > -7. The range of possible solutions for the inequality 3|h| < 21 is all numbers from -7 to 7 (not including -7 and 7).

 

 

 

How about a case where there is more than one term within the absolute value, as in the inequality: |p + 8| > 5? Let’s solve this one too.

 

 

Example

Problem  |p + 8| > 5

P + 8

> 

5

- (p + 8)

> 

5

p + 8 − 8

> 

5 − 8

- (p + 8) • -1

< 

5 • -1

Answer:

p

> 

-3

p + 8 

< 

-5

 

 

 

p + 8 − 8

< 

-5 − 8

 

 

 

Answer:

p

< 

-13

 

For this inequality to be true, we find that p has to be either greater than -3 or less than

-13. This means that the graph of the inequality will be two rays going in opposite directions, as shown below.

 

 

 

Let’s look at one more example: 56 ≥ 7|5 − b|. Since the absolute value term is less than the constant term, we are expecting the solution to be of the “and” sort: a segment between two points on the number line.

 

Example

Problem

56

7|5 − b|

8

|5 − b|

 

8

5 − b

8

- (5 − b)

8 − 5

5 − b − 5

8 • -1

- (5 − b) • -1

3

-b

-8

5 − b

3 • -1

-b • -1

-8 − 5

5 − b − 5

-3

B

-13

-b

Answer:

b

-3

-13 • -1

-b • -1

 

 

 

13

b

 

 

 

Answer:             

b

13

 

 

We find that b ≥ -3 and b ≤ 13, so any point that lies between -3 and 13 (including those points) will be a solution to this problem.

 

 

 

Summary

 

An inequality defines a range of possible values for a mathematical relationship. The range for an absolute value inequality is defined by two possibilities—the original variable may be positive or it may be negative. This is why we have to evaluate it twice, once as a positive term, and once as a negative term.

 

The graph of the solution set of an absolute value inequality will either be a segment between two points on the number line, or two rays going in opposite directions from two points on the number line. Any point along the segment or along the rays will satisfy the original inequality.