1.7: Solve Absolute Value Equations and Inequalities

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1.7 Solve Absolute Value Equations and
Inequalities

• Objectives
• To solve and graph absolute value equations and
  inequalities

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Exercise 1

• Kenny digs a hole in his backyard while his
  parents are still at work. He neatly piles the
  dirt from the hole on the concrete patio. When
  Kennys parents return home, they demand that
  Kenny explain why he destroyed their
  well-manicured backyard oasis. Kenny says that
  the hole and the dirt from the whole demonstrate
  absolute value. What is Kenny talking about?

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Absolute Value

• The absolute value of a number is its distance
  from zero on a real number line.
• The absolute value is always positive.

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Absolute Value Equations
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Solving Absolute Value Equations

• To solve ax b c,
• Write TWO equations
• ax b c
• ax b -c
• Solve each equation.
• Check each solution in the original equation.

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Exercise 2

• Solve each absolute value equation.
• x 15
• 2x 9 15.

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Exercise 3

• Solve.
• 4x 12 28
• 4x 10 6x

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Extraneous Solutions

• In the course of solving an absolute value
  equation, one of the solutions may not actually
  satisfy the original equation. This an
  extraneous solution. Get rid of it its no good!

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Try in your notebooks ?

• Page 55 3-6, 9-15, 21-24, 34-36

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Absolute Value Inequalities

• Follow me here
• x 5 means the distance from zero equals 5.
• x 5 means the distance from zero is less than
  or equal to five.
• -5 x 5

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Absolute Value Inequalities

• Follow me here
• x 5 means the distance from zero equals 5.
• x 5 means the distance from zero is greater
  than or equal to five.
• x -5 or x 5

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Absolute Value Inequalities

• If the general methods from the previous slides
  are incomprehensible, you could just memorize
  these.

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Exercise 4

• Solve the inequality. Then graph the solution.
• x 2 lt 6
• 2x 1 9
• 7 x 4

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Exercise 5

• Solve the inequality. Then graph the solution.
• x 4 6
• 2x 7 gt 1
• 3x 5 10

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Exercise 6

• A professional baseball should weigh 5.125
  ounces, with a tolerance of 0.125 ounces.
  (Tolerance is the maximum deviation from the
  ideal measurement.) Write and solve an absolute
  value inequality that describes the acceptable
  weights of a baseball.

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Solution

• First write a verbal model of the equation
• actual weight- ideal weight tolerance
• Convert verbal model to an equation and solve
• w-5.125 0.125
• -0.125 w-5.125 0.125
• 5 w 5.25

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Assignment

• P. 55 40,43-54,74, 76a-b