Solving Absolute Value Equations

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• Solving Absolute Value Equations

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What is Absolute Value?

• The absolute value of a number is the number of
  units it is from zero on the number line.
• 5 and -5 have the same absolute value.
• The symbol x represents the absolute value of
  the number x.

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• -8 8
• 4 4
• You try
• 15 ?
• -23 ?
• Absolute Value can also be defined as
• if a gt0, then a a
• if a lt 0, then a -a

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We can evaluate expressions that contain absolute
value symbols.

• Think of the bars as grouping symbols.
• Evaluate 9x -3 5 if x -2
• 9(-2) -3 5
• -18 -3 5
• -21 5
• 21 526

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Equations may also contain absolute value
expressions

• When solving an equation, isolate the absolute
  value expression first.
• Rewrite the equation as two separate equations.
• Consider the equation x 3. The equation
  has two solutions since x can equal 3 or -3.
• Solve each equation.
• Always check your solutions.
• Example Solve x 8 3
• x 8 3 and x 8 -3
• x -5 x -11
• Check x 8 3
• -5 8 3 -11 8 3
• 3 3 -3 3
• 3 3 3 3

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Now Try These

• Solve y 4 - 3 0
• y 4 3 You must first
  isolate the variable by adding 3 to both
  sides.
• Write the two separate equations.
• y 4 3 y 4 -3
• y -1 y -7
• Check y 4 - 3 0
• -1 4 -3 0 -7 4 - 3
  0
• -3 - 3 0 -3 - 3 0
• 3 - 3 0 3 - 3 0
• 0 0 0 0

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• 3d - 9 6 0 First isolate the variable by
• subtracting 6 from both sides.
• 3d - 9 -6
• There is no need to go any further with this
  problem!
• Absolute value is never negative.
• Therefore, the solution is the empty set!

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• Solve 3x - 5 12
• x - 5 4
• x - 5 4 and x - 5 -4
  x 9 x 1
• Check 3x - 5 12
• 39 - 5 12 31 - 5 12
• 34 12 3-4 12
• 3(4) 12 3(4) 12
• 12 12 12 12

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• Solve 8 5a 14 - a
• 8 5a 14 - a and 8 5a -(14
  a)
• Set up your 2 equations, but make sure
  to negate the entire right side of the second
  equation.
• 8 5a 14 - a and 8 5a -14 a
• 6a 6 4a -22
  a 1 a -5.5
• Check 8 5a 14 - a
• 8 5(1) 14 - 1 8 5(-5.5) 14 -
  (-5.5)
• 13 13 -19.5
  19.5 13 13 19.5 19.5