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Section 7.2

Section 7.2. Solving Absolute Value Equations. Solving Absolute Value Equations. Def. Absolute value represents the distance a number is from 0. Thus, it is always positive. Absolute value is denoted by the bars |x|. 5 and -5 have the same absolute value.

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Section 7.2

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  1. Section 7.2 Solving Absolute Value Equations

  2. Solving Absolute Value Equations Def. Absolute value represents the distance a number is from 0. Thus, it is always positive. • Absolute value is denoted by the bars |x|. • 5 and -5 have the same absolute value. • The symbol |x| represents the absolute value of the number x. |8| = 8 and |-8| = 8

  3. Examples: • Ex1: Ex2: Ex3:

  4. 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+ 5=26

  5. Solving absolute value equations • First, isolate the absolute value expression. • Set up two equations to solve. • For the first equation, drop the absolute value bars and solve the equation. • For the second equation, drop the bars, negate the opposite side, and solve the equation. • Always check the solutions.

  6. Example: • When solving an equation, isolate the absolute value expression first. • Rewrite the equation as two separate equations. • 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

  7. Examples: • Ex4: Ex5:

  8. Special Notes: • |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!

  9. 6|5x + 2| = 312 • Isolate the absolute value expression by dividing by 6. 6|5x + 2| = 312 |5x + 2| = 52 • Set up two equations to solve. • 5x + 2 = 52 5x + 2 = -52 • 5x = 50 5x = -54 • x = 10 or x = -10.8 • Check:6|5x + 2| = 312 6|5x + 2| = 312 • 6|5(10)+2| = 312 6|5(-10.8)+ 2| = 312 • 6|52| = 312 6|-52| = 312 • 312 = 312 312 = 312

  10. 3|x + 2| -7 = 14 • Isolate the absolute value expression by adding 7 and dividing by 3. 3|x + 2| -7 = 14 3|x + 2| = 21 |x + 2| = 7 • Set up two equations to solve. • x + 2 = 7x + 2 = -7 • x = 5 or x = -9 • Check:3|x + 2| - 7 = 14 3|x + 2| -7 = 14 • 3|5 + 2| - 7 = 14 3|-9+ 2| -7 = 14 3|7| - 7 = 14 3|-7| -7 = 14 • 21 - 7 = 14 21 - 7 = 14 • 14 = 14 14 = 14

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