Holt Algebra 2 2-8 Solving Absolute-Value Equations and Inequalities 2-8,9 Solving and Graphing Absolute-Value Equations and Inequalities Holt Algebra 2 Vocab Vocab Lesson Presentation Lesson Presentation Text Questions Text Questions Objective and PA Math Standards Objective and PA Math Standards Warm Up Warm Up Lesson Plan Lesson Plan Worksheet Worksheet Lesson Quiz Lesson Quiz
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Holt Algebra 2 2-8 Solving Absolute-Value Equations and Inequalities 2-8,9 Solving and Graphing Absolute-Value Equations and Inequalities Holt Algebra.
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Holt Algebra 2
2-8Solving Absolute-Value Equations and Inequalities 2-8,9 Solving and Graphing Absolute-Value
Equations and Inequalities
Holt Algebra 2
VocabVocab
Lesson PresentationLesson Presentation
Text QuestionsText Questions
Objective and PA Math StandardsObjective and PA Math Standards
Warm UpWarm UpLesson PlanLesson Plan
WorksheetWorksheet
Lesson QuizLesson Quiz
Holt Algebra 2
2-8Solving Absolute-Value Equations and Inequalities
Warm UpSolve.1. y + 7 < –11 2. 5 – 2x ≤ 17y < –18 x ≥ –6
Use interval notation to show the graphed numbers
6.
7.
(-2, 3]
(-, 1]
3. f(x) = –|x + 1|
4. f(x) = 2|x| – 1
5. f(x) = |x + 1| + 2
–5; –2
7; 5
7; 4
Evaluate each expression in #3-5 for f(4) and f(-3).
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving Absolute-Value Equations and Inequalities
Solve and graph compound inequalities.
Write, solve and graph absolute-value equations and inequalities.
Objectives
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving Absolute-Value Equations and Inequalities
compound statementdisjunctionconjunctionabsolute-valueabsolute value function
Vocabulary
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving 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.
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving 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< 2 –3 ≤ x < 2
U
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving Absolute-Value Equations and Inequalities
Dis- means “apart.” Disjunctions have two pieces.
Con- means “together” Conjunctions have one piece.
Reading Math
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving 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 positiveabsolute value of any real number is always positive.
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving Absolute-Value Equations and Inequalities
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving Absolute-Value Equations and Inequalities
Example 1A: Solving Compound InequalitiesSolve and graph the compound inequality.
{y|y < –4 or y ≥ –2}
6y < –24 OR y +5 ≥ 3
y < –4 y ≥ –2
–6 –5 –4 –3 –2 –1 0 1 2 3
/6 /6 -5 -5
orInequality Notation
Set Builder Notation
Interval Notation (–∞,-4) U [-2, ∞)
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving Absolute-Value Equations and Inequalities
Example 1C: Solving Compound Inequalities
Solve and graph the compound inequality.
{x|x < 3 or x ≥ 5}.
–3 –2 –1 0 1 2 3 4 5 6
x – 5 < –2 OR –2x ≤ –10
x < 3 or x ≥ 5
+5 +5 /–2 /-2
(–∞, 3) U [5, ∞)
Inequality Notation
Set Builder Notation
Interval Notation
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving Absolute-Value Equations and Inequalities
–4 –3 –2 –1 0 1 2 3 4 5
2x ≥ –6 AND –x > –4
x ≥ –3 x < 4
{x|x ≥ –3 x < 4}. U
[–3, 4)
Solve and graph the compound inequality.
Check It Out! Example 1b
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving Absolute-Value Equations and Inequalities
2 4 6 8 10 12 14 16 18 20
x – 5 < 12 OR 6x ≤ 12
x < 17 x ≤ 2
Note: Every point that satisfies x < 17 also satisfies x < 2 {x|x < 17}.
(-∞, 17)
Solve and graph the compound inequality.
Check It Out! Example 1c
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving 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)
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving Absolute-Value Equations and Inequalities
Example 1B Continued
f(x)
g(x)
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).
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving 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.
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving 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.
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving 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.
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving Absolute-Value Equations and Inequalities
Note: The symbol ≤ can replace <, and the rules still apply. The symbol ≥ can replace >, and the rules still apply.
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving Absolute-Value Equations and Inequalities
Solve the equation.
Example 2A: Solving Absolute-Value Equations
|–3 + k| = 10
–3 + k = 10 or –3 + k = –10
k = 13 or k = –7
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving Absolute-Value Equations and Inequalities
Solve the equation.
Example 2B: Solving Absolute-Value Equations
x = 16 or x = –16
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving Absolute-Value Equations and Inequalities
Check It Out! Example 2a
|x + 9| = 13
Solve the equation.
x + 9 = 13 or x + 9 = –13
x = 4 or x = –22
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving Absolute-Value Equations and Inequalities
Check It Out! Example 2b
|6x| – 8 = 22
Solve the equation.
|6x| = 30
6x = 30 or 6x = –30
x = 5 or x = –5
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving 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”
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving 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 ≥ 8 or –4q ≤ –12
q ≤ –2 or q ≥ 3
{q|q ≤ –2 or q ≥ 3}
(–∞, –2] U [3, ∞)
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving Absolute-Value Equations and Inequalities
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
Check It Out! Example 3a
x < -1 or x > 5
{x|x < –1 or x > 5}
(–∞, –1) U (5, ∞)
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving Absolute-Value Equations and Inequalities
Solve the compound inequality. Then graph the solution set.
Example 4A: Solving Absolute-Value Inequalities with Conjunctions
|2x +7| ≤ 3
2x + 7 ≤ 3 and 2x + 7 ≥ –3
2x ≤ –4 and 2x ≥ –10
x ≤ –2 and x ≥ –5
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving Absolute-Value Equations and Inequalities
Solve the compound inequality. Then graph the solution set.
Example 4B: Solving Absolute-Value Inequalities with Conjunctions
|p – 2| ≤ –6
|p – 2| ≤ –6 and p – 2 ≥ 6
p ≤ –4 and p ≥ 8
Because no real number satisfies both p ≤ –4 andp ≥ 8, there is no solution. The solution set is ø.
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving Absolute-Value Equations and Inequalities
Solve the compound inequality. Then graph the solution set.
|x – 5| ≤ 8
x – 5 ≤ 8 and x – 5 ≥ –8
x ≤ 13 and x ≥ –3
Check It Out! Example 4a
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving Absolute-Value Equations and Inequalities
Solve the compound inequality. Then graph the solution set.
–2|x +5| > 10
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.
Check It Out! Example 4b
|x + 5| < –5
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving Absolute-Value Equations and Inequalities
Lesson Quiz: Part I
Solve. Then graph the solution.
1.
2.
y – 4 ≤ –6 or 2y >8
–7x < 21 and x + 7 ≤ 6
{y|y ≤ –2 ≤ or y > 4}
{x|–3 < x ≤ –1}
–4 –3 –2 –1 0 1 2 3 4 5
–4 –3 –2 –1 0 1 2 3 4 5
Solve each equation.
3. |2v + 5| = 9 4. |5b| – 7 = 13
2 or –7 + 4
Solving and Graphing Absolute-Value Equations and Inequalities
Holt Algebra 2
2-8Solving Absolute-Value Equations and Inequalities
Lesson Quiz: Part II
Solve. Then graph the solution.
5.
6.
7.
|1 – 2x| > 7 {x|x < –3 or x > 4}
|3k| + 11 > 8 R
–2|u + 7| ≥ 16 ø
–4 –3 –2 –1 0 1 2 3 4 5
–4 –3 –2 –1 0 1 2 3 4 5
Solving and Graphing Absolute-Value Equations and Inequalities