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2-8 Solving Absolute-Value Equations and Inequalities Objectives Solve compound inequalities. Write and solve absolute-value equations and inequalities. Holt Algebra 2 2-8 Solving Absolute-Value Equations and Inequalities 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 A disjunction is true if and only if at least one of its parts is true. Holt Algebra 2 2-8 Solving Absolute-Value Equations and Inequalities A conjunction is a compound statement that uses the word and. Conjunction: x ≥ –3 AND 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 Holt Algebra 2 2-8 Solving Absolute-Value Equations and Inequalities Example 1A: Solving Compound Inequalities Solve the compound inequality. Then graph the solution set. 6y < –24 OR y +5 ≥ 3 Solve both inequalities for y. 6y < –24 y + 5 ≥3 or y < –4 y ≥ –2 The solution set is all points that satisfy {y < –4 or y ≥ –2}. –6 –5 –4 –3 –2 –1 Holt Algebra 2 0 1 2 3 (–∞, –4) U [–2, ∞) 2-8 Solving Absolute-Value Equations and Inequalities Example 1B: Solving Compound Inequalities Solve the compound inequality. Then graph the solution set. Solve both inequalities for c. and 2c + 1 < 1 c ≥ –4 c<0 The solution set is the set of points that satisfy both c ≥ –4 and c < 0. [–4, 0) –6 –5 –4 –3 –2 –1 Holt Algebra 2 0 1 2 3 2-8 Solving Absolute-Value Equations and Inequalities Example 1C: Solving Compound Inequalities Solve the compound inequality. Then graph the solution set. x – 5 < –2 OR –2x ≤ –10 Solve both inequalities for x. x – 5 < –2 or –2x ≤ –10 x<3 x≥5 The solution set is the set of all points that satisfy {x < 3 or x ≥ 5}. –3 –2 –1 Holt Algebra 2 0 1 2 3 4 5 6 (–∞, 3) U [5, ∞) 2-8 Solving Absolute-Value Equations and Inequalities Check It Out! Example 1b Solve the compound inequality. Then graph the solution set. 2x ≥ –6 AND –x > –4 Solve both inequalities for x. 2x ≥ –6 and –x > –4 x ≥ –3 x<4 The solution set is the set of points that satisfy both {x ≥ –3 x < 4}. –4 –3 –2 –1 0 Holt Algebra 2 [–3, 4) 1 2 3 4 5 2-8 Solving Absolute-Value Equations and Inequalities 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. Holt Algebra 2 2-8 Solving 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. Holt Algebra 2 2-8 Solving 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. Holt Algebra 2 2-8 Solving 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. Holt Algebra 2 2-8 Solving Absolute-Value Equations and Inequalities Helpful Hint Think: Greator inequalities involving > or ≥ symbols are disjunctions. Think: Less thand inequalities involving < or ≤ symbols are conjunctions. Holt Algebra 2 2-8 Solving Absolute-Value Equations and Inequalities Note: The symbol ≤ can replace <, and the rules still apply. The symbol ≥ can replace >, and the rules still apply. Holt Algebra 2 2-8 Solving Absolute-Value Equations and Inequalities Example 2A: Solving Absolute-Value Equations Solve the equation. |–3 + k| = 10 This can be read as “the distance from k to –3 is 10.” –3 + k = 10 or –3 + k = –10 Rewrite the absolute value as a disjunction. k = 13 or k = –7 Add 3 to both sides of each equation. Holt Algebra 2 2-8 Solving Absolute-Value Equations and Inequalities Check It Out! Example 2a Solve the equation. This can be read as “the distance from x to +9 is 4.” |x + 9| = 13 x + 9 = 13 or x + 9 = –13 x=4 Holt Algebra 2 or x = –22 Rewrite the absolute value as a disjunction. Subtract 9 from both sides of each equation. 2-8 Solving Absolute-Value Equations and Inequalities Check It Out! Example 2b Solve the equation. |6x| – 8 = 22 Isolate the absolutevalue expression. |6x| = 30 6x = 30 or 6x = –30 x=5 Holt Algebra 2 or x = –5 Rewrite the absolute value as a disjunction. Divide both sides of each equation by 6. 2-8 Solving Absolute-Value Equations and Inequalities You can solve absolute-value inequalities using the same methods that are used to solve an absolute-value equation. Holt Algebra 2 2-8 Solving 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 q ≤ –2 Holt Algebra 2 Rewrite the absolute value as a disjunction. or –4q ≤ –12 Subtract 2 from both sides of each inequality. or q ≥ 3 Divide both sides of each inequality by –4 and reverse the inequality symbols. 2-8 Solving Absolute-Value Equations and Inequalities Example 3A Continued {q ≤ –2 or q ≥ 3} (–∞, –2] U [3, ∞) –3 –2 –1 0 1 2 3 4 5 6 To check, you can test a point in each of the three region. |–4(–3) + 2| ≥ 10 |14| ≥ 10 Holt Algebra 2 |–4(0) + 2| ≥ 10 |2| ≥ 10 x |–4(4) + 2| ≥ 10 |–14| ≥ 10 2-8 Solving 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 Rewrite the absolute value as a disjunction. 4x > 20 or 4x < –4 Add 8 to both sides of each inequality. x>5 Holt Algebra 2 or x < –1 Divide both sides of each inequality by 4. 2-8 Solving Absolute-Value Equations and Inequalities Check It Out! Example 3a Continued {x < –1 or x > 5} (–∞, –1) U (5, ∞) –3 –2 –1 0 1 2 3 4 5 6 To check, you can test a point in each of the three region. |4(–2) + 8| > 12 |–16| > 12 Holt Algebra 2 |4(0) + 8| > 12 |8| > 12 x |4(6) + 8| > 12 |32| > 12 2-8 Solving Absolute-Value Equations and Inequalities Check It Out! Example 3b Solve the inequality. Then graph the solution. |3x| + 36 > 12 Isolate the absolute value as a disjunction. |3x| > –24 3x > –24 or x > –8 or 3x < 24 Rewrite the absolute value as a disjunction. x<8 Divide both sides of each inequality by 3. The solution is all real numbers, R. (–∞, ∞) –3 –2 –1 Holt Algebra 2 0 1 2 3 4 5 6 2-8 Solving 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 Multiply both sides by 3. 2x + 7 ≤ 3 and 2x + 7 ≥ –3 Rewrite the absolute value as a conjunction. 2x ≤ –4 and x ≤ –2 and Holt Algebra 2 2x ≥ –10 Subtract 7 from both sides of each inequality. x ≥ –5 Divide both sides of each inequality by 2. 2-8 Solving Absolute-Value Equations and Inequalities Example 4A Continued The solution set is {–5 ≤ x ≤ 2}. –6 –5 –3 –2 –1 Holt Algebra 2 0 1 2 3 4 2-8 Solving Absolute-Value Equations and Inequalities Check It Out! Example 4a Solve the compound inequality. Then graph the solution set. |x – 5| ≤ 8 Multiply both sides by 2. x – 5 ≤ 8 and x – 5 ≥ –8 Rewrite the absolute value as a conjunction. x ≤ 13 and Add 5 to both sides of each inequality. Holt Algebra 2 x ≥ –3 2-8 Solving Absolute-Value Equations and Inequalities Check It Out! Example 4 The solution set is {–3 ≤ x ≤ 13}. –10 Holt Algebra 2 –5 0 5 10 15 20 25 2-8 Solving 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 Divide both sides by –2, and reverse the inequality symbol. Rewrite the absolute value as a conjunction. Subtract 5 from both sides of each inequality. Because no real number satisfies both x < –10 and x > 0, there is no solution. The solution set is ø. Holt Algebra 2