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6-5 Solving Open Sentences
  Involving Absolute Value
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                                                                1
Objectives
• Students will solve absolute value equations.
• Students will solve absolute value
  inequalities.




                                                  2
Absolute Value Equations
|x|=5 means the distance between x and 0 is 5
    units, so x = 5 or –5.
When solving equations, there are 2 cases to
    consider:
1. The value inside the absolute value signs is
    positive.
2. The value inside the absolute value signs is
    negative.

                                                  3
Example 1
|x + 4| = 5
Case 1: x + 4 = 5
           x=1
Case 2: x + 4 = -5
          x = -9




                           4
Example 2
|x – 7| = 8
Case 1: x – 7 = 8
           x = 15
Case 2: x – 7 = -8
          x = -1




                           5
Writing Absolute Value Equations
• Find the point that is in the middle of the 2
  points.
• Find the distance between the midpoint and
  the endpoints.                         distance from midpoint
                                         to endpoints
      •
               •           |x - 6| = 3
      3           9
                                   midpoint




                                                           6
Solving Absolute Value
             Inequalities
• Case 1: The value inside the absolute value
  signs is greater than the given value of n.
• Case 2: The value inside the absolute value
  signs is less than the opposite of n.
• If |x|<n, then x<n and x>-n “within” “and”
• If |x|>n, then x>n and x<-n “outside” “or”



                                                7
Example 3
|3y – 3| > 9
3y – 3 > 9 or 3y – 3 < -9
 3y > 12 or 3y < -6
 y>4         or y < -2


   -2           4



                                8
Example 4
|s - 3| ≤ 12
s – 3 ≤ 12     and        s – 3 ≥ -12
s ≤ 15         and        s ≥ -9


    -9               15




                                        9

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Algebra Solving Open Sentences Involving Absolute Value

  • 1. 6-5 Solving Open Sentences Involving Absolute Value This presentation was created following the Fair Use Guidelines for Educational Multimedia. Certain materials are included under the Fair Use exemption of the U. S. Copyright Law. Further use of these materials and this presentation is restricted. 1
  • 2. Objectives • Students will solve absolute value equations. • Students will solve absolute value inequalities. 2
  • 3. Absolute Value Equations |x|=5 means the distance between x and 0 is 5 units, so x = 5 or –5. When solving equations, there are 2 cases to consider: 1. The value inside the absolute value signs is positive. 2. The value inside the absolute value signs is negative. 3
  • 4. Example 1 |x + 4| = 5 Case 1: x + 4 = 5 x=1 Case 2: x + 4 = -5 x = -9 4
  • 5. Example 2 |x – 7| = 8 Case 1: x – 7 = 8 x = 15 Case 2: x – 7 = -8 x = -1 5
  • 6. Writing Absolute Value Equations • Find the point that is in the middle of the 2 points. • Find the distance between the midpoint and the endpoints. distance from midpoint to endpoints • • |x - 6| = 3 3 9 midpoint 6
  • 7. Solving Absolute Value Inequalities • Case 1: The value inside the absolute value signs is greater than the given value of n. • Case 2: The value inside the absolute value signs is less than the opposite of n. • If |x|<n, then x<n and x>-n “within” “and” • If |x|>n, then x>n and x<-n “outside” “or” 7
  • 8. Example 3 |3y – 3| > 9 3y – 3 > 9 or 3y – 3 < -9 3y > 12 or 3y < -6 y>4 or y < -2 -2 4 8
  • 9. Example 4 |s - 3| ≤ 12 s – 3 ≤ 12 and s – 3 ≥ -12 s ≤ 15 and s ≥ -9 -9 15 9