Lesson 7 More or Less Absolute Practice Understanding

Jump Start

Determine which inequality matches each phrase.

$x > 13$
$x \geq 13$
$x < 13$
$x \leq 13$

a.

No more than $13$

b.

A minimum of $13$

c.

More than $13$

d.

No less than $13$

e.

At least $13$

f.

A maximum of $13$

g.

Cannot exceed $13$

h.

Fewer than $13$

Learning Focus

Solve inequalities that contain absolute value.

Represent the solutions to absolute value inequalities with number lines, verbal descriptions, and mathematical notation.

What does it mean to take the absolute value of a number?

What does absolute value describe when it is used in an inequality?

Open Up the Math: Launch, Explore, Discuss

What does absolute value mean?

Since $| 3 | = 3$ and $| - 3 | = 3$ , many people think of absolute value as an operation that makes numbers positive. That’s a useful interpretation to start with, but let’s expand our understanding a bit so that we can apply it to more situation. Consider the following questions.

1.

a.

What numbers make this equation true?

$| x | = 5$

b.

Plot the solutions on the number line.

c.

What do you notice about the solutions?

2.

a.

What numbers make this equation true?

$| x | = 8$

b.

Plot the solutions on the number line.

c.

What do you notice about the solutions?

From this pattern, some people describe an equation like $| x | = 3$ to mean “the numbers that are $3$ units from $0$ on the number line”.

3.

Using your understanding of absolute value and distance, write the absolute value equation this has these solutions:

$x = 11, x = - 11$

4.

Let’s think about inequalities with absolute value.

a.

Describe in words what this absolute value inequality might mean:

$| x | < 3$

b.

Use your interpretation of the inequality to graph the solutions on the number line and write the solutions using mathematical notation.

Solutions:

c.

How can you test your solutions to see if they are correct?

Now you have some absolute value inequality puzzles to be solved using your new interpretation of absolute value. As you work, you’ll find that the inequalities get a little trickier. Look for patterns and try to develop a strategy for solving absolute value inequalities that can be used even when the inequality gets complicated.

5.

Absolute Value Inequality
Verbal Description of the Absolute Value Inequality	All the numbers that are less than or equal to $5$ units from $0$ on the number line.
Number Line Graph of Solutions (Note: All number lines should have arrows at both ends.)
Solutions Written in Mathematical Notation	$- 5 \leq x \leq 5$

6.

Absolute Value Inequality	$\| x \| > 3$
Verbal Description of the Absolute Value Inequality	All the numbers that are greater than $3$ units away from $0$ on the number line.
Number Line Graph of Solutions (Note: All number lines should have arrows at both ends.)
Solutions Written in Mathematical Notation	$x < - 3$ or $x > 3$

7.

Absolute Value Inequality	$\| x \| \geq 6$
Verbal Description of the Absolute Value Inequality
Number Line Graph of Solutions (Note: All number lines should have arrows at both ends.)
Solutions Written in Mathematical Notation

8.

Absolute Value Inequality	$\| x - 5 \| < 3$
Verbal Description of the Absolute Value Inequality
Number Line Graph of Solutions (Note: All number lines should have arrows at both ends.)
Solutions Written in Mathematical Notation	$2 < x < 8$

9.

Absolute Value Inequality
Verbal Description of the Absolute Value Inequality	All the numbers that are less than or equal to $2$ units from $- 4$ on the number line.
Number Line Graph of Solutions (Note: All number lines should have arrows at both ends.)
Solutions Written in Mathematical Notation	$- 6 \leq x \leq - 2$

10.

Absolute Value Inequality	$\| x + 2 \| \geq 6$
Verbal Description of the Absolute Value Inequality
Number Line Graph of Solutions (Note: All number lines should have arrows at both ends.)
Solutions Written in Mathematical Notation

11.

Absolute Value Inequality
Verbal Description of the Absolute Value Inequality
Number Line Graph of Solutions (Note: All number lines should have arrows at both ends.)
Solutions Written in Mathematical Notation	$x < - 9$ or $x > - 3$

12.

Absolute Value Inequality	$\| 2 x - 1 \| \geq 5$
Verbal Description of the Absolute Value Inequality
Number Line Graph of Solutions (Note: All number lines should have arrows at both ends.)
Solutions Written in Mathematical Notation

13.

Absolute Value Inequality	$\| 5 x + 3 \| < 23$
Verbal Description of the Absolute Value Inequality
Number Line Graph of Solutions (Note: All number lines should have arrows at both ends.)
Solutions Written in Mathematical Notation

Ready for More?

Here are the trickiest absolute value inequalities of all. See if you can solve them using the strategies you developed in the earlier problems.

1.

$3 | 2 x - 5 | + 7 \leq 28$

2.

$- 5 | 3 x + 4 | + 12 \leq - 3$

Takeaways

Inequality:	Description:
$\| 3 x + 4 \| > 6$	Given

Inequality:	Description:
$\| 4 x + 5 \| \leq 7$	Given

Vocabulary

absolute value
compound inequality
Bold terms are new in this lesson.

Lesson Summary

In this lesson we learned to solve inequalities with absolute value. We learned that the solutions can be represented on the number line, sometimes with a single interval, and other times with two separate intervals. The solutions are written using mathematical notation as compound inequalities.

Retrieval

Determine which of the coordinate pairs are solutions to the equation. Provide justification by showing your work.

1.

$7 x - 2 y = 28$

A.

$(0, - 7)$

B.

$(8, 14)$

C.

$(4, 0)$

D.

$(- 2, - 9)$

2.

$y = - 3 x + 12$

A.

$(0, 12)$

B.

$(8, - 12)$

C.

$(14, - 2)$

D.

$(- 2, 18)$

3.

Solve each equation for the indicated letter. Show your work as justification for your solution.

$5 x + 3 y = 30$ solve for $x$ .