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**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

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**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.

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**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.

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**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

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**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.

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**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

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**Solving and Graphing Absolute-Value Equations and Inequalities**

Dis- means “apart.” Disjunctions have two pieces. Con- means “together” Conjunctions have one piece. Reading Math

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**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.

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**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.

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**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 –

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**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 –

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**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 –

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**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)

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**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)

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**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)

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**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.

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**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.

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**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.

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**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.

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**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

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**Solving and Graphing Absolute-Value Equations and Inequalities**

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

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**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

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**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

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**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”

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**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, ∞)

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**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, ∞)

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**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

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**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.

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**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

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**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.

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**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

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**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

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