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Lesson 6 - Absolute Value

This document covers the topic of solving equations and inequalities involving absolute value. It defines absolute value both algebraically and informally, provides examples of solving absolute value equations and inequalities, and explains the methods for determining solution sets. Additionally, it includes practice problems for further exploration of the concepts presented.

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34 views3 pages

Lesson 6 - Absolute Value

This document covers the topic of solving equations and inequalities involving absolute value. It defines absolute value both algebraically and informally, provides examples of solving absolute value equations and inequalities, and explains the methods for determining solution sets. Additionally, it includes practice problems for further exploration of the concepts presented.

Uploaded by

superhero1040
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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A2TH

Section 1.6: Equations and Inequalities Involving Absolute Value


In today’s lesson, we will extend our solving skills to include equations involving absolute value.
Algebraic Definition of Absolute Value: Informal Definition of Absolute Value:
Absolute Value is the distance from a
|𝑥| = $ 𝑥 if 𝑥 ≥ 0 number to zero. This distance is always
−𝑥 if 𝑥 < 0 positive!

Examples:
|4| =

|0| =

|−4| = |−4| = |4| =

Solving Absolute Value Equations


Since the absolute value of 𝑥 and −𝑥 have the same value, we need to be careful when solving
absolute value equations. The following statement will help you to solve absolute value equations:

If 𝑎 is a positive real number and 𝑢 is any algebraic expression, then the solutions to the equation

|𝑢| = 𝑎 are ___________ OR _____________

Example 1
Solve the following absolute value equations.
(a) |𝑐 − 15| = 23 (b) |5 − 𝑥| + 8 = 1 (c) |10 − 2𝑥| = 4

Exploration
Use the solving method above to solve each inequality. Graph your solutions on the number line.
(a) |𝑥| > 6 (b) |𝑥| ≤ 6
What did you find?

Absolute Value and Inequalities


If |𝒙| > 𝒂 , then the solutions are: 𝒙 > 𝒂 or −𝒙 > 𝒂
(“or” implies union of 2 sets)

If |𝒙| < 𝒂 , then the solutions are: 𝒙 < 𝒂 and −𝒙 < 𝒂


(“and” implies intersection of 2 sets)

Example 2
Solve the following absolute value inequalities and express your answer using interval notation.
Graph the solution set.

(a) |𝑥 − 15| < 23 (b) |−2𝑥 + 1| ≥ 9


(c) |−𝑤 + 3| − 5 ≥ 2 (d) |3 − 4𝑥| < −4

You Try!
If |𝑥 − 5| < 7, find values for 𝑎 and 𝑏 that make 𝑎 < 3𝑥 − 4 < 𝑏.

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