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Absolute Values Solving Absolute Value Equations

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10/2/2013 Absolute Value Equations 2 General Procedure Form is: for constants a, b and k ≥ 0 By definition: Absolute Value Equations either | a x + b | = a x + b OR ( a x + b) = –k | a x + b | = k = k a x + b = k + – | a x + b | = –( a x + b) Thus

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10/2/2013 Absolute Value Equations 3 Example 1 Solve –3x – 2 = 5 –3x – 2 = 5 or –3x – 2 = –5 –3x = 5 + 2 or –3x = –5 + 2 x = –7/3 or x = 1 Solution set is: { –7/3, 1 } Absolute Value Equations

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10/2/2013 Absolute Value Equations 4 Example 2 Solve 3x = –6 Note: 3x ≥ 0 for all x Hence, 3x = –6 is FALSE for every value of x Thus there is no solution and thus solution set is Absolute Value Equations O or { } Note: WHY ? O ≠ O { }

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10/2/2013 Absolute Value Equations 5 Example 3 Solve 2x – 3 – 4 = 7 Absolute Value Equations 2x – 3 = 7 + 4 = 11 2x – 3 = 11 or –(2x – 3 ) = 11 2x – 3 = 11 or 2x – 3 = –11 2x = 14 or 2x = –8 x = 7 or x = –4 Solution set is { –4, 7 }

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10/2/2013 Absolute Value Equations 6 Example 4 Solve 9 + x = 3 – 2x Removing one absolute value at a time: 9 + x = 3 – 2x and 9 + x = –(3 – 2x) Absolute Value Equations –(9 + x) = –(3 – 2x)9 + x = 3 – 2x –(9 + x) = 3 – 2x 9 + x = –(3 – 2x) Same 9 + x = 3 – 2x 9 + x = –(3 – 2x)

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10/2/2013 Absolute Value Equations 7 9 + x = 3 – 2x Solve 9 + x = 3 – 2x Absolute Value Equations 9 + x = 3 – 2x 9 + x = –(3 – 2x) Solution set is { –2, 12 } or x = 12 or 12 = x or 9 + 3 = –x + 2x or 9 + x = –3 + 2x 2x + x = 3 – 9 3x = –6 x = –2

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10/2/2013 Absolute Value Equations 8 Think about it !

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