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2-8 Solving Absolute-Value Equations and Inequalities Who uses this? Absolute value can be used to represent the acceptable ranges for the dimensions of baseball bats classified by length or weight. (See Exercise 43.) Objectives Solve compound inequalities. Write and solve absolutevalue equations and inequalities. Vocabulary disjunction conjunction absolute value A compound statement is made up of more than one equation or inequality. A disjunction is a compound statement that uses the word or. n È { Ó Dis- means “apart.” Disjunctions have two separate pieces. Con- means “together.” Conjunctions represent one piece. ä Ó { È n ⎧ ⎫ Disjunction: x ≤ -3 OR x > 2 Set builder notation: ⎨x|x ≤-3 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. n È { Ó ä Ó { È n ⎧ ⎫ Conjunction: x ≥ -3 AND x < 2 Set builder notation: ⎨x|x ≥ -3 x < 2⎬ ⎩ Conjunctions ⎭can be A conjunction is true if and only if all of its parts are true. written as a single statement as shown. x ≥ -3 and x < 2 → -3 ≤ x < 2 EXAMPLE 1 Solving Compound Inequalities Solve each compound inequality. Then graph the solution set. A x + 3 ≤ 2 OR 3x > 9 Solve both inequalities for x. x+3≤2 or x ≤ -1 3x > 9 x>3 ⎧ ⎫ The solution set is all points that satisfy ⎨x | x ≤ -1 or x > 3⎬. ⎩ ⎭ n È { Ó ä Ó { È n (-∞, -1] (3, ∞) B -2x < 8 AND x - 3 ≤ 2 Solve both inequalities for x. -2x < 8 and x-3≤2 x > -4 x≤5 The solution set is the set of points that satisfy both x > -4 and x ≤ 5, ⎧ ⎫ ⎨ x | x - 4 < x ≤ 5⎬ ⎩ ⎭ n È { Ó 150 ä Ó { È n (-4, 5] Chapter 2 Linear Functions a207se_c02l08_0150_0156.indd 150 1/10/06 2:34:34 PM Solve each compound inequality. Then graph the solution set. C x + 3 > 7 OR 3x ≥ 18 Solve both inequalities for x. x+3>7 or 3x ≥ 18 x>4 x≥6 Because every point that satisfies x ≥ 6 also satisfies x > 4, the solution ⎧ ⎫ set is ⎨ x | x > 4⎬. ⎩ ⎭ { Ó ä Ó { È (4, ∞) n Solve each compound inequality. Then graph the solution set. 1a. x - 2 < 1 or 5x ≥ 30 1b. 2x ≥ -6 and -x > -4 1c. x - 5 < 12 or 6x ≤ 12 1d. -3x < -12 and x + 4 ≤ 12 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 WORDS NUMBERS The absolute value of a real number x, ⎪x⎥, is equal to its distance from zero on a number line. ALGEBRA ⎪5⎥ = 5 ⎧ x if x ≥ 0 ⎨ -x if x < 0 ⎩ ⎪x⎥ = ⎪-5⎥ = 5 Absolute-value equations and inequalities can be represented by compound statements. Consider the equation ⎪x⎥ = 3. Think: Greator inequalities involving > or ≥ symbols are disjunctions. Think: Less thand inequalities involving < or ≤ symbols are conjunctions. 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. ȟÝȟÊÊÎ x { Î Ó £ ä £ Ó Î { x Ó Î { x Ó Î { x ȟÝȟÊÊÎ x { Î Ó £ ä £ ȟÝȟÊÊÎ x { Î Ó £ ä £ Absolute-Value Equations and Inequalities For all real numbers x and all positive real numbers a: ⎪x⎥ = a ⎪x⎥ < a x = -a OR x = a x >-a AND x < a ⎪x⎥ > a x < -a OR x > a -a < x < a Note: The symbol ≤ can replace <, and the rules still apply. The symbol ≥ can replace >, and the rules still apply. 2- 8 Solving Absolute-Value Equations and Inequalities a207se_c02l08_0150_0156.indd 151 151 12/14/05 2:55:35 PM EXAMPLE 2 Solving Absolute-Value Equations Solve each equation. A ⎪x - 7⎥ = 5 This can be read as “the distance from x to 7 is 5.” x - 7 = 5 or x - 7 = -5 Rewrite the absolute value as a disjunction. x = 12 or x = 2 Add 7 to both sides of each equation. B ⎪3x⎥ + 5 = 14 ⎪3x⎥ = 9 Isolate the absolute-value expression. 3x = 9 or 3x = -9 x = 3 or x = -3 Rewrite the absolute value as a disjunction. Divide both sides of each equation by 3. Solve each equation. 2a. ⎪x + 9⎥ = 13 2b. ⎪6x⎥ - 8 = 22 You can solve absolute-value inequalities using the same methods that are used to solve an absolute-value equation. Solving an Absolute-value Inequality 1. Isolate the absolute-value expression, if necessary. 2. Rewrite the absolute-value expression as a compound inequality. 3. Solve each part of the compound inequality for x. EXAMPLE 3 Solving Absolute-Value Inequalities with Disjunctions Solve each inequality. Then graph the solution set. A ⎪2x + 1⎥ > 5 In Example 3B, if you recognize that 2x + 1 > 5 or 2x + 1 < -5 Rewrite the absolute value as a disjunction. 2x > 4 or 2x < -6 Subtract 1 from both sides of each inequality. x > 2 or x < -3 ⎫ ⎧ ⎨ x | x > 2 x < -3 ⎬ ⎭ ⎩ Divide both sides of each inequality by 2. n È { Ó ⎪expression⎥ > -8 ä Ó { È (-∞, -3) (2, ∞) n To check, you can test a point in each of the three regions. ⎪2(-4) + 1⎥ > 5 ⎪2(0) + 1⎥ > 5 ⎪2(5) + 1⎥ > 5 ⎪1⎥ > 5 ✘ ⎪11⎥ > 5 ✔ ⎪-7⎥ > 5 ✔ is always true, you will know the solution immediately. B ⎪4x⎥ + 16 > 8 ⎪4x⎥ > -8 Isolate the absolute-value expression. 4x > -8 or 4x < 8 Rewrite the absolute value as a disjunction. x > -2 or x < 2 Divide both sides of each inequality by 4. { Ó ä Ó { È n (-∞, ∞) The solution set is all real numbers, . Solve each inequality. Then graph the solution set. 3a. ⎪4x - 8⎥ > 12 152 3b. ⎪3x⎥ + 36 > 12 Chapter 2 Linear Functions a207se_c02l08_0150_0156.indd 152 12/14/05 2:55:38 PM EXAMPLE 4 Solving Absolute-Value Inequalities with Conjunctions Solve each inequality. Then graph the solution set. A ⎪3x - 9⎥ _ ≤ 12 2 ⎪3x - 9⎥ ≤ 24 Multiply both sides by 2. 3x - 9 ≤ 24 and 3x - 9 ≥ -24 Rewrite the absolute value as a conjunction. 3x ≤ 33 and 3x ≥ -15 x ≤ 11 and x ≥ -5 Add 9 to both sides of each inequality. Divide both sides of each inequality by 3. ⎧ ⎫ The solution set is ⎨x |-5 ≤ x ≤ 11⎬. ⎩ ⎭ È { Ó In Example 4B, if you recognize that ⎪expression⎥ ≤ -2 is never true, you will know the solution immediately. ä Ó { È n £ä £Ó B -4 ⎪x + 3⎥ ≥ 8 ⎪x + 3⎥ ≤ -2 Divide both sides by -4, and reverse the inequality symbol. x + 3 ≤ -2 and x + 3 ≥ 2 Rewrite the absolute value as a conjunction. x ≤ -5 and x ≥ -1 Subtract 3 from both sides of each inequality. Because no real number satisfies both x ≤ -5 and x ≥ -1, there is no solution. The solution set is ∅. Solve each inequality. Then graph the solution set. ⎪x - 5⎥ 4a. _ ≤ 4 2 4b. -2 ⎪x + 5⎥ > 10 THINK AND DISCUSS 1. Explain why the solution set to ⎪7x⎥ > -1 is all real numbers. 2. Explain why there is no solution to ⎪x + 3⎥ ≤ -2. Give another example of an absolute-value equation that has no solution. 3. Write an absolute-value inequality to model “the distance between x and 5 is greater than 10.” 4. GET ORGANIZED Copy and complete the graphic organizer. Use the flowchart to explain the decisions and steps needed to solve an absolute-value equation or inequality. µÕ>ÌÊÀ iµÕ>ÌÞ¶ µÕ>Ì ÃÕVÌÊÀ VÕV̶ ÃÕVÌ ÕVÌ 2- 8 Solving Absolute-Value Equations and Inequalities a207se_c02l08_0150_0156.indd 153 153 12/14/05 2:55:40 PM