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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

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

Slide 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.

Slide 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 ( ) ( )

Slide 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.

Slide 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

Slide 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)

Slide 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.

Slide 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.

Slide 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.

Slide 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 [ ) [ )

Slide 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,  )

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

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

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

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

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

Slide 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.

Slide 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.

Slide 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.

Slide 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.

Slide 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.

Slide 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.

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

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

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

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

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

Slide 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).

Slide 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) ( )

Slide 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.

Slide 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.

Slide 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 x <  12 A number line solution: Set-builder notation: {x| x  2} Interval notation: ( ,  12)  (  2,  ) ) (

Slide 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: ( ,  )

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

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

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

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