1. Solving and Graphing Absolute-Value Equations and Inequalities 2-8,9
Lesson Plan
Warm Up
Objective and PA Math Standards
Vocab
Lesson Presentation
Text Questions
Worksheet
Lesson Quiz
Holt Algebra 2

2. Solving and Graphing Absolute-Value Equations and Inequalities
Warm Up
Solve.
1. y + 7 < –11
y < –18
2. 5 – 2x ≤ 17
x ≥ –6
Evaluate each expression in #3-5 for f(4) and f(-3).
3. f(x) = –|x + 1|
–5; –2
4. f(x) = 2|x| – 1
7; 5
5. f(x) = |x + 1| + 2
7; 4
Use interval notation to show the graphed numbers 6.
(-2, 3]
(-, 1] 7.

3. Objectives Solve and graph compound inequalities.
Solving and Graphing Absolute-Value
Equations and Inequalities
Objectives
Solve and graph compound inequalities.
Write, solve and graph absolute-value equations and inequalities.

4. Vocabulary compound statement disjunction conjunction absolute-value
Solving and Graphing Absolute-Value
Equations and Inequalities
Vocabulary
compound statement
disjunction
conjunction
absolute-value
absolute value function

5. Solving and Graphing Absolute-Value Equations and Inequalities
A compound statement is made up of more than one equation or inequality.
A disjunction is a compound statement using or.
Disjunction: x ≤ –3 OR x > 2 Set builder notation: {x|x ≤ –3 U x > 2}
Interval Notation: (-∞, -2] U (2, ∞)
A disjunction is true if at least one of its parts is true.

6. Solving and Graphing Absolute-Value Equations and Inequalities
A conjunction is a compound statement using and.
Conjunction: x ≥ –3 AND x < 2 Set builder notation: {x|x ≥ –3 x < 2}
Interval Notation: [-3, 2)
A conjunction is true only if all of its parts are true. Conjunctions can be written as a single statement as shown.
x ≥ –3 and x< –3 ≤ x < 2 U

7. Solving and Graphing Absolute-Value Equations and Inequalities
Dis- means “apart.” Disjunctions have two pieces.
Con- means “together” Conjunctions have one piece.
Reading Math

8. Solving and Graphing Absolute-Value Equations and Inequalities
Absolute value of a number x, written |x|, is the distance from x to zero on the number line. Because it represents distance without regard to direction, the absolute value of any real number is always positive.

9. Solving and Graphing Absolute-Value Equations and Inequalities
An absolute-value function is composed of two linear pieces, one with a negative slope and one with a positive slope.
The graph of the absolute-value “parent” function f(x) = |x| (shown above) has a V shape with its vertex at (0, 0) and slopes of -1 and 1.

10. Solving and Graphing Absolute-Value Equations and Inequalities
Example 1A: Solving Compound Inequalities
Solve and graph the compound inequality.
6y < –24 OR y +5 ≥ 3
/ /
Inequality Notation
Set Builder Notation
Interval Notation
y < –4 or
y ≥ –2
{y|y < –4 or y ≥ –2}
(–∞,-4) U [-2, ∞)
–6 –5 –4 –3 –2 –

11. Solving and Graphing Absolute-Value Equations and Inequalities
Example 1C: Solving Compound Inequalities
Solve and graph the compound inequality.
x – 5 < –2 OR –2x ≤ –10
/– /-2
Inequality Notation
Set Builder Notation
Interval Notation
x < 3 or x ≥ 5
{x|x < 3 or x ≥ 5}.
(–∞, 3) U [5, ∞)
–3 –2 –

12. Solving and Graphing Absolute-Value Equations and Inequalities
Check It Out! Example 1b
Solve and graph the compound inequality.
2x ≥ –6 AND –x > –4
x ≥ – x < 4
{x|x ≥ – x < 4}. U
[–3, 4)
–4 –3 –2 –

13. Solving and Graphing Absolute-Value Equations and Inequalities
Check It Out! Example 1c
Solve and graph the compound inequality.
x – 5 < 12 OR 6x ≤ 12
x < x ≤ 2
Note: Every point that satisfies x < 17 also satisfies x < {x|x < 17}.
(-∞, 17)

14. Solving and Graphing Absolute-Value Equations and Inequalities
Example 1A Continued
The graph of g(x) = |x|– 5 is the graph of f(x) = |x| after a vertical shift of 5 units down. The vertex of g(x) is (0, –5).
f(x)
g(x)

15. Solving and Graphing Absolute-Value Equations and Inequalities
Example 1B Continued
The graph of g(x) = |x + 1| is the graph of f(x) = |x| after a horizontal shift of 1 unit left. The vertex of g(x) is (–1, 0).
f(x)
g(x)

16. Solving and Graphing Absolute-Value Equations and Inequalities
Absolute-value equations and inequalities can be represented by compound statements. Consider the equation |x| = 3.
The solutions of |x| = 3 are the two points that are 3 units from zero. The solution is a disjunction: x = –3 or x = 3.

17. Solving and Graphing Absolute-Value Equations and Inequalities
The solutions of |x| < 3 are the points that are less than 3 units from zero. The solution is a conjunction: –3 < x < 3.

18. Solving and Graphing Absolute-Value Equations and Inequalities
The solutions of |x| > 3 are the points that are more than 3 units from zero. The solution is a disjunction:
x < –3 or x > 3.

19. Solving and Graphing Absolute-Value Equations and Inequalities
Note: The symbol ≤ can replace <, and the rules still apply. The symbol ≥ can replace >, and the rules still apply.

20. Solving and Graphing Absolute-Value Equations and Inequalities
Example 2A: Solving Absolute-Value Equations
Solve the equation.
|–3 + k| = 10
–3 + k = 10 or –3 + k = –10
k = 13 or k = –7

21. Solving and Graphing Absolute-Value Equations and Inequalities
Example 2B: Solving Absolute-Value Equations
Solve the equation.
x = 16 or x = –16

22. Solving and Graphing Absolute-Value Equations and Inequalities
Check It Out! Example 2a
Solve the equation.
|x + 9| = 13
x + 9 = 13 or x + 9 = –13
x = 4 or x = –22

23. Solving and Graphing Absolute-Value Equations and Inequalities
Check It Out! Example 2b
Solve the equation.
|6x| – 8 = 22
|6x| = 30
6x = 30 or 6x = –30
x = 5 or x = –5

24. Remember this: “LESS THAN AND… GREATER THAN OR.”
Solving and Graphing Absolute-Value
Equations and Inequalities
You can solve absolute-value inequalities using the same methods that are used to solve an absolute-value equation.
Remember this:
“LESS THAN AND…
GREATER THAN OR.”
When the abs. value inequality is <, it’s an “AND”
When the abs. value inequality is >, it’s an “OR”

25. Solving and Graphing Absolute-Value Equations and Inequalities
Example 3A: Solving Absolute-Value Inequalities with Disjunctions
Solve the inequality. Then graph the solution.
|–4q + 2| ≥ 10
–4q + 2 ≥ 10 or –4q + 2 ≤ –10
–4q ≥ or –4q ≤ –12
q ≤ –2 or q ≥ 3
{q|q ≤ –2 or q ≥ 3}
(–∞, –2] U [3, ∞)

26. Solving and Graphing Absolute-Value Equations and Inequalities
Check It Out! Example 3a
Solve the inequality. Then graph the solution.
|4x – 8| > 12
4x – 8 > 12 or 4x – 8 < –12
4x > 20 or 4x < –4
x > 5 or x < –1
x < -1 or x > 5
{x|x < –1 or x > 5}
(–∞, –1) U (5, ∞)

27. Solving and Graphing Absolute-Value Equations and Inequalities
Example 4A: Solving Absolute-Value Inequalities with Conjunctions
Solve the compound inequality. Then graph the solution set.
|2x +7| ≤ 3
2x + 7 ≤ 3 and 2x + 7 ≥ –3
2x ≤ –4 and 2x ≥ –10
x ≤ –2 and x ≥ –5

28. Solving and Graphing Absolute-Value Equations and Inequalities
Example 4B: Solving Absolute-Value Inequalities with Conjunctions
Solve the compound inequality. Then graph the solution set.
|p – 2| ≤ –6
|p – 2| ≤ –6 and p – 2 ≥ 6
p ≤ –4 and p ≥ 8
Because no real number satisfies both p ≤ –4 and p ≥ 8, there is no solution. The solution set is ø.

29. Solving and Graphing Absolute-Value Equations and Inequalities
Check It Out! Example 4a
Solve the compound inequality. Then graph the solution set.
|x – 5| ≤ 8
x – 5 ≤ 8 and x – 5 ≥ –8
x ≤ 13 and x ≥ –3

30. Solving and Graphing Absolute-Value Equations and Inequalities
Check It Out! Example 4b
Solve the compound inequality. Then graph the solution set.
–2|x +5| > 10
|x + 5| < –5
x + 5 < –5 and x + 5 > 5
x < –10 and x > 0
Because no real number satisfies both x < –10 and x > 0, there is no solution. The solution set is ø.

31. Solving and Graphing Absolute-Value Equations and Inequalities
Lesson Quiz: Part I
Solve. Then graph the solution. 1.
y – 4 ≤ –6 or 2y >8
{y|y ≤ –2 ≤ or y > 4}
–4 –3 –2 – 2.
–7x < 21 and x + 7 ≤ 6
{x|–3 < x ≤ –1}
–4 –3 –2 –
Solve each equation.
3. |2v + 5| = 9
4. |5b| – 7 = 13
2 or –7
+ 4

32. Solving and Graphing Absolute-Value Equations and Inequalities
Lesson Quiz: Part II
Solve. Then graph the solution. 5.
|1 – 2x| > 7
{x|x < –3 or x > 4}
–4 –3 –2 – 6.
|3k| + 11 > 8 R
–4 –3 –2 – 7.
–2|u + 7| ≥ 16 ø