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Solving Absolute Value Equations _ Inequalities

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									 Solving Absolute Value
Equations & Inequalities
  What is this used for?
• You can use absolute value
  equations and inequalities to help
  with real-life problems, such as
  finding acceptable weights and
  tolerances for manufactured
  products and sports equipment.
  What is an absolute value?
• An absolute value of a number is its
  distance away from zero on a number
  line. Since we measure distances with
  positive numbers, the absolute value of
  a number is a positive value.
• The absolute value of -2 is 2, and the
  absolute value of 2 is 2.



    -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9
  How do I solve an absolute
      value equation?
• Rewrite the equation as it is, only
  without the absolute value
  symbols. Solve it.
• Write a 2nd equation, changing the
  sign of the term(s) on the right
  side. Solve it.
             ax + b = c

•   ax + b = c       ax + b = - c



• Set up both equations, and solve
  each one. You will have 2
  possible solutions.
  How do I solve an absolute
      value inequality?
• Rewrite the inequality as it is, only
  without the absolute value
  symbols. Solve it.
• Write a 2nd inequality, changing
  the sign of the term(s) on the right
  side and switching the inequality
  symbol around. Solve it.
             ax + b < c

•   ax + b < c   and   ax + b > - c



• Set up both inequalities, and solve
  each one. You will have 2
  possible solutions.
               ax + b >c

•   ax + b >c     or   ax + b < - c



• Set up both inequalities, and
  solve each one. You will have
    2 possible solutions.
    Real-Life Examples
A cereal manufacturer has a tolerance of 0.75
  ounce for a box of cereal that is supposed to
  weigh 20 ounces. Write and solve an
  absolute value inequality that describes the
  acceptable weights for “20 ounce” boxes.
         Real-Life Examples:
          Cereal, continued
It’s the absolute value of the actual weight minus the
   ideal weight that equals the tolerance.
   x – 20 < 0.75         Rewrite this like so:
                       x – 20 < 0.75 and x – 20 > -0.75

                          Solve them both.
           Real-Life Examples:
            Cereal, continued
It’s the absolute value of the actual weight minus the ideal
    weight that equals the tolerance.
   x – 20 < 0.75         Rewrite this like so:
                       x – 20 < 0.75 and x – 20 > - 0.75
                         + 20 + 20           + 20 + 20


                           x < 20.75 and x > 19.25
                            So, 19.25 < x < 20.75
Tolerances in
Manufacturing
   A manufacturer has a 0.6
   tolerance for a bottle of
   salad dressing advertised
   as 16 oz. Write and solve
   an absolute value
   inequality that describes
   the acceptable volumes
   for “16 oz” bottles.
Tolerances in
Manufacturing
 x – 16 < 0.6


x – 16 < 0.6 and x – 16 > - 0.6
Tolerances in
Manufacturing
 x – 16 < 0.6
means that:
x – 16 < 0.6 and x – 16 > -0.6
x < 16.6   and   x > 15.4
x is anything between those,
so:   15.4 < x < 16.6
            Quality Control
• You are a quality control inspector at a
  bowling pin company. A regulation bowling
  pin must weigh between 50 ounces and 58
  ounces, inclusive. Write an absolute value
  inequality describing the weights you should
  accept and another one to describe the
  weights you should reject.
            Quality Control
• Let the weight of the pin = w.
• Find the average of the extreme
  weights:
        50 + 58 = 54
           2
This means that the tolerance is
 + 4 pounds. So, the inequality is:
 w – 54 < 4. You would accept
  these.
You should reject a bowling pin if
  its weight satisfies w - 54 > 4.
         Sports Equipment


Sport        Wt. Range   72.Volleyball:

Volleyball   260-280 g     v – 270 < 10

Basketball   600-650 g
Water Polo   400-450 g
Lacrosse     142-149 g
Football     14-15 oz.
         Sports Equipment


Sport        Wt. Range   72.Volleyball:

Volleyball   260-280 g     v – 270 < 10

Basketball   600-650 g     Basketball:
Water Polo   400-450 g    b – 625 < 25
Lacrosse     142-149 g
Football     14-15 oz.
         Sports Equipment


Sport        Wt. Range   72.Volleyball:

Volleyball   260-280 g     v – 270 < 10

Basketball   600-650 g     Basketball:
Water Polo   400-450 g    b – 625 < 25
Lacrosse     142-149 g     Water Polo:
Football     14-15 oz.    w - 425 < 25
         Sports Equipment

                         72.Volleyball:

Sport        Wt. Range     v – 270 < 10

Volleyball   260-280 g     Basketball:
Basketball   600-650 g    b – 625 < 25
Water Polo   400-450 g     Water Polo:
Lacrosse     142-149 g    w - 425 < 25
Football     14-15 oz.     Lacrosse:
                           l – 145.5 < 3.5
         Sports Equipment
                         72.Volleyball:
                           v – 270 < 10
                           Basketball:
Sport        Wt. Range
                          b – 625 < 25
Volleyball   260-280 g
                           Water Polo:
Basketball   600-650 g
                          w - 425 < 25
Water Polo   400-450 g
                           Lacrosse:
Lacrosse     142-149 g
                           l – 145.5 < 3.5
Football     14-15 oz.
                           Football:
                           f – 14.5 < 0.5
         Sports Equipment
                         73. Volleyball:
                           v – 270 > 10
                           Basketball:
Sport        Wt. Range
                          b – 625 > 25
Volleyball   260-280 g
                           Water Polo:
Basketball   600-650 g
                          w - 425 > 25
Water Polo   400-450 g
                           Lacrosse:
Lacrosse     142-149 g
                           l – 145.5 > 3.5
Football     14-15 oz.
                           Football:
                           f – 14.5 > 0.5

								
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