2-8 Solving Absolute-Value Equations and Inequalities Warm Up Lesson Presentation Lesson Quiz Holt McDougal Algebra 2 Holt Algebra 2

Opener-SAME SHEET-5/11 Solve. 1. y + 7 < –11 y < –18 2. 4m ≥ –12 3. 5 – 2x ≤ 17 x ≥ –6 Use interval notation to indicate the graphed numbers. 4. (-2, 3] (-, 1] 5.

Objectives Solve compound inequalities. Write and solve absolute-value equations and inequalities.

Vocabulary disjunction conjunction absolute-value

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A compound statement is made up of more than one equation or inequality. A disjunction is a compound statement that uses the word or. Disjunction: x ≤ –3 OR x > 2 Set builder notation: {x|x ≤ –3 U x > 2} A disjunction is true if and only if at least one of its parts is true.

A conjunction is a compound statement that uses the word and. Conjunction: x ≥ –3 AND x < 2 Set builder notation: {x|x ≥ –3 x < 2}. A conjunction is true if and 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

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

Example 1A: Solving Compound Inequalities Solve the compound inequality. Then graph the solution set. 6y < –24 OR y +5 ≥ 3 The solution set is all points that satisfy {y|y < –4 or y ≥ –2}. –6 –5 –4 –3 –2 –1 0 1 2 3 (–∞, –4) U [–2, ∞)

Example 1B: Solving Compound Inequalities Solve the compound inequality. Then graph the solution set. The solution set is the set of points that satisfy both c ≥ –4 and c < 0. –6 –5 –4 –3 –2 –1 0 1 2 3 [–4, 0)

Example 1C: Solving Compound Inequalities Solve the compound inequality. Then graph the solution set. x – 5 < –2 OR –2x ≤ –10 The solution set is the set of all points that satisfy {x|x < 3 or x ≥ 5}. –3 –2 –1 0 1 2 3 4 5 6 (–∞, 3) U [5, ∞)

Solve the compound inequality. Then graph the solution set. Check It Out! Example 1a Solve the compound inequality. Then graph the solution set. x – 2 < 1 OR 5x ≥ 30 The solution set is all points that satisfy {x|x < 3 U x ≥ 6}. –1 0 1 2 3 4 5 6 7 8 (–∞, 3) U [6, ∞)

Solve the compound inequality. Then graph the solution set. Check It Out! Example 1d Solve the compound inequality. Then graph the solution set. –3x < –12 AND x + 4 ≤ 12 The solution set is the set of points that satisfy both {x|4 < x ≤ 8}. 2 3 4 5 6 7 8 9 10 11 (4, 8]

Solve the compound inequality. Then graph the solution set. Check It Out! Example 1b Solve the compound inequality. Then graph the solution set. 2x ≥ –6 AND –x > –4 The solution set is the set of points that satisfy both {x|x ≥ –3 x < 4}. U –4 –3 –2 –1 0 1 2 3 4 5 [–3, 4)

Opener-SAME SHEET-10/11 Let g(x) be the indicated transformation of f(x) = 2x + 7. Write the rule for g(x). 1. horizontal translation 4 units left 2. reflection across the x-axis 3. Horizontal stretch by a factor of 4.

Solve. Then graph the solution. 2-8 Hmwk Quiz A Solve. Then graph the solution. 1. y – 4 ≤ –6 or 2y >8 –4 –3 –2 –1 0 1 2 3 4 5 2. –7x < 21 and x + 7 ≤ 6 –4 –3 –2 –1 0 1 2 3 4 5

Recall that the absolute value of a number x, written |x|, is the distance from x to zero on the number line. Because absolute value represents distance without regard to direction, the absolute value of any real number is nonnegative.

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.

The solutions of |x| < 3 are the points that are less than 3 units from zero. The solution is a conjunction: –3 < x < 3.

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.

Helpful Hint Think: Greator inequalities involving > or ≥ symbols are disjunctions. Think: Less thand inequalities involving < or ≤ symbols are conjunctions.

Note: The symbol ≤ can replace <, and the rules still apply Note: The symbol ≤ can replace <, and the rules still apply. The symbol ≥ can replace >, and the rules still apply.

Example 2A: Solving Absolute-Value Equations Solve the equation. |–3 + k| = 10 |x + 9| = 13 |6x| – 8 = 22

You can solve absolute-value inequalities using the same methods that are used to solve an absolute-value equation.

Example 3A: Solving Absolute-Value Inequalities with Disjunctions Solve the inequality. Then graph the solution. |0.5r| – 3 ≥ –3 |–4q + 2| ≥ 10 q ≤ –2 or q ≥ 3 The solution is all real numbers, R. –3 –2 –1 0 1 2 3 4 5 6

Example 4A: Solving Absolute-Value Inequalities with Conjunctions Solve the compound inequality. Then graph the solution set. |4x – 8| > 12 x ≤ –2 and x ≥ –5 (–∞, –1) U (5, ∞) –3 –2 –1 0 1 2 3 4 5 6 –6 –5 –3 –2 –1 0 1 2 3 4

Solve the compound inequality. Then graph the solution set. Check It Out! Example 4a Solve the compound inequality. Then graph the solution set. Because no real number satisfies both p ≤ –4 and p ≥ 8, there is no solution. The solution set is ø. x ≤ 13 and x ≥ –3 –10 –5 0 5 10 15 20 25

Solve. Then graph the solution. Opener-SAME SHEET-5/19 NEED BOOKS Solve. Then graph the solution. 1. |1 – 2x| > 7 {x|x < –3 or x > 4} –4 –3 –2 –1 0 1 2 3 4 5 2. x ≤ 13 and x ≥ –3 –10 –5 0 5 10 15 20 25

6. |3k| + 11 > 8 R –4 –3 –2 –1 0 1 2 3 4 5 7. –2|u + 7| ≥ 16 ø 33

Review Quiz

Review Quiz

Review Quiz

Book Problems P. 154 #1-13 and #28-31 and #36

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