2-8 Solving Absolute-Value Equations and Inequalities

1 150 Chapter 2 Linear FunctionsA compound statement is made up of more than one equation or disjunction is a compound statement that uses the word or. { n 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 conjunction is a compound statement that uses the word and. { n n { 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 < 21 EXAMPLE Solving Compound InequalitiesSolve each compound inequality. Then graph the solution x + 3 2 OR 3x > 9 Solve both Inequalities for + 3 2 or 3x > 9 x -1 x > 3 The solution set is all points that satisfy x | x -1 or x > 3 . { n n { (- , -1] (3, )B -2x < 8 AND x - 3 2 Solve both Inequalities for < 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 n { (-4, 5] 2-8 Solving Absolute-Value Equations and InequalitiesObjectivesSolve compound and solve Absolute-Value Equations and valueWho uses this?))}}}}}}}}

2 absolute value can be used to represent the acceptable ranges for the dimensions of baseball bats classified by length or weight. (See Exercise 43.)Dis- means apart. Disjunctions have two separate means together. Conjunctions represent one 1501/10/06 2:34:34 PM1/10/06 2:34:34 PM {x x { {x x { {x x { 2- 8 Solving Absolute-Value Equations and Inequalities151 Solve each compound inequality. Then graph the solution x + 3 > 7 OR 3x 18 Solve both Inequalities for + 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 . { n { (4, ) 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.}}}}}}}}

3 Because absolute value represents distance without regard to direction, the absolute value of any real number is absolute value of a real number x, x , is equal to its distance from zero on a number line. 5 = 5 -5 = 5 x if x 0 x = -x if x < 0 absolute ValueAbsolute- value Equations and Inequalities can be represented by compound statements. Consider the equation x = solutions of x = 3 are the two points that are 3 units from zero. The solution is a disjunction: x = -3 or x = solutions of x < 3 are the points that are less than 3 units from zero. The solution is a conjunction: -3 < x < solutions of x > 3 are the points that are more than 3 units from zero. The solution is a disjunction: x < -3 or x > all real numbers x and all positive real numbers a: x = ax = -a OR x = a x < ax >-a AND x < a-a < x < a x > ax < -a OR x > a Absolute-Value Equations and InequalitiesNote: The symbol can replace <, and the rules still apply.

4 The symbol can replace >, and the rules still : Greatorinequalities involving > or symbols are : Less thandinequalities involving < or symbols are 15112/14/05 2:55:35 PM12/14/05 2:55:35 PM152 Chapter 2 Linear Functions2 EXAMPLE Solving Absolute-Value EquationsSolve each 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 3x + 5 = 14 3x = 9 Isolate the Absolute-Value = 9 or 3x = -9 Rewrite the absolute value as a disjunction. x = 3 or x = -3 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 Inequality1. Isolate the Absolute-Value expression, if Rewrite the Absolute-Value expression as a compound Solve each part of the compound inequality for Solving Absolute-Value Inequalities with DisjunctionsSolve each inequality.

5 Then graph the solution 2x + 1 > 52x + 1 > 5 or 2x + 1 < -5 Rewrite the absolute value as a > 4 or 2x < -6 Subtract 1 from both sides of each > 2 or x < -3 Divide both sides of each inequality by 2. x | x > 2 x < -3 { n n { (- , -3) (2, )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 -7 > 5 1 > 5 11 > 5 B 4x + 16 > 8 4x > -8 Isolate the Absolute-Value > -8 or 4x < 8 Rewrite the absolute value as a > -2 or x < 2 Divide both sides of each inequality by 4.{ n { (- , )The solution set is all real numbers, . Solve each inequality. Then graph the solution set. 3a. 4x - 8 > 12 3b. 3x + 36 > 12In Example 3B, if you recognize that expression > -8is always true, you will know the solution 15212/14/05 2:55:38 PM12/14/05 2:55:38 PM2- 8 Solving Absolute-Value Equations and Inequalities1534 EXAMPLE Solving Absolute-Value Inequalities with ConjunctionsSolve each inequality.}}}}

6 Then graph the solution 3x - 9 _ 2 12 3x - 9 24 Multiply both sides by - 9 24 and 3x - 9 -24 Rewrite the absolute value as a conjunction. 3x 33 and 3x -15 Add 9 to both sides of each inequality. x 11 and x -5 Divide both sides of each inequality by solution set is x |-5 x 11 . n { { B -4 x + 3 8 x + 3 -2 Divide both sides by -4, and reverse the inequality + 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 no real number satisfies both x -5 and x -1, there is no solution. The solution set is . Solve each inequality.}}

7 Then graph the solution set. 4a. x - 5 _ 2 4 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 > i > V V V V V > Copy and complete the graphic organizer. Use the flowchart to explain the decisions and steps needed to solve an Absolute-Value equation or Example 4B, if you recognize that expression -2is never true, you will know the solution 15312/14/05 2:55:40 PM12/14/05 2:55:40 PM