3.5: Solving Equations and Inequalities with Absolute Value

Absolute Value

Example 1

Find the distance between the following two points on the real number line.

[ ] [ ] [ ] [ ]

Definition 1. The distance between two real numbers is the absolute value of their ____________________________.

Example 2

Rewrite the absolute value statements as distance statements.

[ ] is the same as saying “The distance between ______ and ______.”

[ ] is the same as saying “The distance between ______ and ______.”

[ ] is the same as saying “The distance between ______ and ______.”

[ ] is the same as saying “The distance between ______ and ______.”

Example 3

You’re currently on Highway-75 at mile marker 44. You read a sign that says, “Next exit: 14 miles.”

[A] Draw the scenario, letting x be the two possible exits.

[B] What are the two possible exits? (What is x equal to?)

[C] How can we write a mathematical equation for this visual representation? Recall the definition of absolute

value.

Example 4 From p. 287, #4, 7, 10

[ ]

is the same as saying “The distance between _______ and ______ is equal to _____.” Now, draw and solve.

[ ]

is the same as saying “The distance between _______ and ______ is equal to _____.” Now, draw and solve.

[ ] is the same as saying “The distance between _______ and ______ is equal to _____.” Now, draw and solve.

Example 5

[A] [B]

Inequalities with Absolute Value

We will solve inequalities with absolute value almost just like we solve equations with absolute value! The key

again is to DRAW IT OUT. Just like with equations involving absolute value, begin by isolating the absolute value

expression.

Example 6

Solve and graph the solution set on a number line:

[ ] [ ]

[ ]

Example 7 From p. 288, #61

Synthesis

Solve for