Warm-Up

• Evaluate each expression, given that x=3 and y=-2. a. |2x -9|

Answer: 1) -3 2) 3 3) 15 4) -15

b. |y –x| Answer: 1) -5 2) 1 3) -1 4) 5

• Solve.

|3x + 6| = 9Answer:1) x=1, -5 2) x= -1, 5 3) x= 3, -15 4) x= -3, 15

6.4: Absolute Values and Inequalities

Objective:

•Learn how to solve absolute value inequalities.

Review

• Why is the absolute value of a number always greater than or equal to zero?

• Two or more inequalities connected by the words _______ or _________ are a compound inequality.

Conjunction: |ax + b| < c

Means: x is between + c

-c < ax +b < cLess Than when an absolute value is on the left and the inequality symbol is < or ≤, the compound sentence uses and.

Disjunction: |ax +b| > c

Means: not between!

ax + b < -c or ax + b > c

Greater Than when an absolute value is on the left and the inequality symbol is > or ≥, the compound sentence uses or.

Solving absolute inequalities and graphing:

|x - 4| < 3 (less than is between)

Means: -3 < x- 4 < 3 (solve)

Graph:

+4 +4 +4

1< x< 7

0 1 2 3 4 5 6 7 8 9

Solving absolute inequalities and

graphing:• |s – 3| ≤ 12 (less than is between) Means: -12 ≤ s – 3 ≤ 12 (solve) + 3 + 3 + 3

- 9 ≤ s ≤ 15

Graph:

-9 -6 -3 0 3 6 9 12 15 18 21 24

Check Your Progress

• Solve each absolute value inequalities then graph.

• A. |y + 4| < 5

• B. |z – 3| ≤ 2

Solve and graph:

|x + 9 |> 13 (disjunction)

Means: x + 9 < -13 or x + 9 > 13 -

9-9 -9 -9

x < -22

x > 4Graph:

-25 -20 -15 -10 -5 0 5 10

Check Your Progress

• Solve each absolute value inequalities and graph.

• A. | 3y – 3| > 9

• B. |2x + 7| ≥ 11

Change the graph to an absolute value inequality:

1. Write the inequality. (x is between)

2 < x < 8

2. Find half way between 2 and 8 It’s 5 (this is the median)

To find the median, add the two numbers and then divide by 2. 2+8 = 5

0 1 2 3 4 5 6 7 8 9 10

2

3. Now rewrite the inequality and subtract 5 (the median) from each section.

2 - 5 < x - 5 < 8 - 5

Combine like terms or numbers and you get -3 < x - 5 < 3

4. Write your absolute inequality|x - 5| < 3

Notice: The median is 3 units away from either number.

Write the inequality for this disjunction:

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

1.x < -6 or x > 4 (find the median)

2. x + 1 < - 5 x+1 > 5

3. |x+1|>5

+1 +1(subtract -1 from both sides, so add 1)

+1 +1

(write x + 1 inside the absolute brackets and 5 outside positive)

Check Your Progress

Write an absolute value inequality for the graph shown

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8

Closing the lesson:

• Summarize the major points of the lesson and answer the Essential Question: How are absolute value inequalities like linear inequalities?

Homework:

• Textbook page 316 #8-30 even, 31 – 36, 38 –40