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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Linear Equations and Inequalities in One Variable CHAPTER 8.1 Compound Inequalities 8.2 Equations Involving Absolute Value 8.3 Inequalities Involving Absolute Value 8.4 Functions and Graphing 8.5 Function Operations 8

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Compound Inequalities 8.1 1.Solve compound inequalities involving “and”. 2.Solve compound inequalities involving “or”.

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Slide 8 - 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Compound Inequality: Two inequalities joined by either “and” or “or.” Examples:x > 3 and x 8 2 x or x > 4 Intersection: For two sets A and B, the intersection of A and B, symbolized by A B, is a set containing only elements that are in both A and B.

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Slide 8 - 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example For the compound inequality x > 5 and x < 2, graph the solution set and write the compound inequality without “and,” if possible. Then write in set-builder notation and in interval notation. Solution The set is the region of intersection. x > 5 x < 2 x > 5 and x < 2 ( ) ( )

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Slide 8 - 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued x > 5 and x < 2 Without “and”: 5 < x < 2 Set-builder notation: {x| 5 < x < 2} Interval notation: ( 5, 2) Warning: Be careful not to confuse the interval notation with an ordered pair.

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Slide 8 - 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example For the inequality graph the solution set. Then write the solution set in set-builder notation and in interval notation. Solution Solve each inequality in the compound inequality. and

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Slide 8 - 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued [ ) [ ) Without “and”: 2 x < 4 Set-builder notation: {x| 2 x < 4} Interval notation: [ 2, 4)

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Slide 8 - 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Compound Inequalities Involving “and” To solve a compound inequality involving “and”, 1. Solve each inequality in the compound inequality. 2. The solution set will be the intersection of the individual solution sets.

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Slide 8 - 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution For the compound inequality, graph the solution set. Then write the solution set in set-builder notation and in interval notation. Since no number is greater than 5 and less than 1, the solution set is the empty set 0 Example Set builder notation: { } or Interval notation: We do not write interval notation because there are no values in the solution set.

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Slide 8 - 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Union: For two sets A and B, the union of A and B, symbolized by A B, is a set containing every element in A or in B. Solving Compound Inequalities Involving “or” To solve a compound inequality involving “or”, 1. Solve each inequality in the compound inequality. 2. The solution set will be the union of the individual solution sets.

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Slide 8 - 12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example For the compound inequality, graph the solution set. Then write the solution set in set-builder notation and in interval notation. Solution or [ ) [ )

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Slide 8 - 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley continued Solution set: Set-builder notation: {x|x < 1 or x 1} Interval notation: ( , 1) [1, )

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Slide 8 - 14 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve: a) b) c) d)

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Slide 8 - 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve: a) b) c) d)

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Slide 8 - 16 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve: a) b) c) d)

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Slide 8 - 17 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve: a) b) c) d)

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Equations Involving Absolute Value 8.2 1.Solve equations involving absolute value.

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Slide 8 - 19 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Absolute Value Property If |x| = a, where x is a variable or an expression and a 0, then x = a or x = a.

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Slide 8 - 20 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solve. a. |2x +1| = 5 b. |3 – 4x| = –10 Solution a. b. |3 – 4x| = –10 The solution set is {–3, 2}. x = –3 or x = 2 2x = –6 or 2x = 4 2x +1 = –5 or 2x +1 = 5 This equation has the absolute value equal to a negative number. Because the absolute value of every real number is a positive number or zero, this equation has no solution.

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Slide 8 - 21 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Equations Containing a Single Absolute Value To solve an equation containing a single absolute value, 1. Isolate the absolute value so that the equation is in the form |ax + b| = c. If c > 0, proceed to steps 2 and 3. If c < 0, the equation has no solution. 2. Separate the absolute value into two equations, ax + b = c and ax + b = c. 3. Solve both equations.

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Slide 8 - 22 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solve. |3x + 4| + 3 = 11 Solution |3x + 4| + 3 = 11 Subtract 3 from both sides to isolate the absolute value. |3x + 4| = 8 3x + 4 = 8 or 3x + 4 = –8 3x = 4 or 3x = –12 x = 4/3 or x = –4 The solutions are 4/3 and 4.

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Slide 8 - 23 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Equations in the Form |ax + b| = |cx + d| To solve an equation in the form |ax + b| = |cx + d|, 1. Separate the absolute value equation into two equations ax + b = cx + d, and ax + b = (cx + d). 2. Solve both equations.

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Slide 8 - 24 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Solve: |3x – 5| = |8 + 4x|. Solution 3x – 5 = 8 + 4x or 3x – 5 = (8 + 4x) –13 + 3x = 4x –13 = x The solutions are 13 and 3/7.

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Slide 8 - 25 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve: a) b) c) d)

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Slide 8 - 26 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve: a) b) c) d)

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Slide 8 - 27 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve: a) b) c) all real numbers d) no solution

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Slide 8 - 28 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve: a) b) c) all real numbers d) no solution

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Inequalities Involving Absolute Value 8.3 1.Solve absolute value inequalities involving less than. 2.Solve absolute value inequalities involving greater than.

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Slide 8 - 30 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Inequalities in the Form |x| 0 1. Rewrite as a compound inequality involving “and”: x > a and x < a. (We can also use a < x < a.) 2. Solve the compound inequality. Similarly, to solve |x| a, we would write x a and x a (or a x a).

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Slide 8 - 31 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example For the inequality, solve, graph the solution set, and write the solution set in both set-builder and interval notation. |x – 3| < 6. Solution |x – 3| < 6 6 < x – 3 < 6 Rewrite as a compound inequality. 3 < x < 9 Add 3 to each part of the inequality. A number line solution: Set-builder notation: {x| 3 < x < 9} Interval notation ( 3, 9). 10-9-7-5-313579-10-8-4048-10-26-6102 ( )

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Slide 8 - 32 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example For the inequality, solve, graph the solution set, and write the solution set in both set-builder and interval notation. |2x – 3| + 8 < 5. Solution Isolate the absolute value. |2x – 3| + 8 < 5 |2x – 3| < –3 Since the absolute value cannot be less than a negative number, this inequality has no solution: . Set builder notation: { } or Interval notation: We do not write interval notation because there are no values in the solution set.

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Slide 8 - 33 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Inequalities in the Form |x| > a, where a > 0 1. Rewrite as a compound inequality involving “or”: x a. 2. Solve the compound inequality. Similarly, to solve |x| a, we would write x a or x a.

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Slide 8 - 34 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example For the inequality, solve, graph the solution set, and write the solution set in both set-builder and interval notation. |x + 7| > 5. Solution We convert to a compound inequality and solve each. |x + 7| > 5 x + 7 5 x < 12 A number line solution: Set-builder notation: {x| x 2} Interval notation: ( , 12) ( 2, ). 5-14-12-10-8-6-4-2024-15-13-9-53-15-71-115-3 ) (

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Slide 8 - 35 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example For the inequality, solve, graph the solution set, and write the solution set in both set-builder and interval notation. |4x + 7| 9 > 12. Solution Isolate the absolute value|4x + 7| 9 > 12 |4x + 7| > 3 This inequality indicates that the absolute value is greater than a negative number. Since the absolute value of every real number is either positive or 0, the solution set is . Set-builder notation: {x|x is a real number} Interval notation: ( , ). 5-14-12-10-8-6-4-2024-15-13-9-53-15-71-115-3

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Slide 8 - 36 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve: a) b) c) d)

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Slide 8 - 37 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve: a) b) c) d)

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Slide 8 - 38 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve: a) b) c) d)

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Slide 8 - 39 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve: a) b) c) d)

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