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Warm-Up

Warm-Up. Solve for the indicated variable. x + y = 2 for y 2x + 2y = -8 for x 2x + 7y = 14 for y 6x - 2y = 9 for x Simplify. 5. -2 ⁵ 6. (-3)² 7. 5³ 8. -6². -32. 9. 125. -36. Solving Absolute Value Equations. Absolute value is denoted by the bars |3|.

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Warm-Up

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  1. Warm-Up • Solve for the indicated variable. • x + y = 2 for y • 2x + 2y = -8 for x • 2x + 7y = 14 for y • 6x - 2y = 9 for x • Simplify. • 5. -2⁵ • 6. (-3)² • 7. 5³ • 8. -6² -32 9 125 -36

  2. Solving Absolute Value Equations

  3. Absolute value is denoted by the bars |3|. Absolute value represents the distance a number is from 0. Thus, it is always positive. |8| = 8 and |-8| = 8 Solving Absolute Value Equations

  4. Absolute-Value Equations WORDS NUMBERS The equations |x| = a asks, what values of x have an absolute value of a? The solutions are a and the opposite of a. |x| = 5 x = 5 or x = -5 GRAPHS ALGEBRA ←a units→←a units→ |x| = a x = a or x = -a (a ≥ 0)   -a 0 a ABSOLUTE VALUE CANNOT EQUAL A NEGATIVE ex: |x| = -5 NO SOLUTION!!! Distance CANNOT be Negative!!!!!

  5. SOLVING ABSOLUTE-VALUE EQUATIONS You can solve some absolute-value equations using mental math. For instance, the equation |x| 8 has two solutions: 8 and 8 because both 8 and -8 are 8 units from zero. Note: Ask yourself “What numbers are 8 units from 0?” To solve absolute-value equations, you can use the fact that the expression inside the absolute value symbols can be either positive or negative.

  6. 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. Solving Absolute Value Equations

  7. Solving an Absolute-Value Equation The expressionx  2 can be equal to 5 or 5. x  2IS NEGATIVE | x  2 |  5 x  2IS POSITIVE x  2 IS POSITIVE x  2 IS POSITIVE x  2 IS NEGATIVE x  2  5 | x  2 |  5 | x  2 |  5 x  3 x  2  5 x  2  5 x  2  5 x  2  5 CHECK x  7 x  7 Solve | x  2 |  5 Solve | x  2 |  5 SOLUTION The expressionx  2can be equal to5or5. x  2 IS POSITIVE x  2 IS NEGATIVE | x  2 |  5 | x  2 |  5 x  2  5 x  2  5 x  7 x  3 The equation has two solutions: 7 or –3. | 7  2 |  | 5 |  5 |3  2 |  | 5 |  5

  8. Solving an Absolute-Value Equation Isolate theabsolute value expressionon one side of the equation. 2x  7 IS NEGATIVE 2x  7 IS POSITIVE 2x  7 IS NEGATIVE 2x  7 IS POSITIVE 2x  7 IS POSITIVE | 2x  7 |  5  4 | 2x  7 |  5  4 | 2x  7 |  5  4 | 2x  7 |  9 | 2x  7 |  9 | 2x  7 |  9 2x  7  9 2x  7  +9 2x  7  9 2x  7  +9 2x  7  +9 2x  2 2x  16 2x  16 x  8 x  1 x  1 TWO SOLUTIONS x  8 x  8 Solve | 2x  7 |  5  4 Solve | 2x  7 |  5  4 SOLUTION Isolate theabsolute value expressionon one side of the equation. 2x  7 IS POSITIVE 2x  7 IS NEGATIVE | 2x  7 |  5  4 | 2x  7 |  5  4 | 2x  7 |  9 | 2x  7 |  9 2x  7  +9 2x  7  9 2x  16 2x  2 x  8 x  1

  9. Solving an Absolute-Value Equation Isolate theabsolute value expressionon one side of the equation. x  5 IS NEGATIVE x  5 IS POSITIVE 3| x  5 |  18 3| x  5 | 18 | x  5 |  6 | x  5 |  6 x  5  6 x  5  +6 +5 +5 x  11 x  1 x  1 TWO SOLUTIONS x  11 Solve 3|x  5 |  18 SOLUTION Isolate theabsolute value expressionon one side of the equation. +5 +5

  10. Isolate the absolute value expression by dividing by 6. 6|5x + 2| = 312 |5x + 2| = 52 Set up two equations to solve. 6|5x + 2| = 312 • 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

  11. 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. 3|x + 2| -7 = 14 • 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

  12. Solving an Absolute-Value Equation Recall that |x |is the distance between x and 0. If |x | = 8, then the solution can be either 8 or 8 is a solution of the equation.  8  7  6  5  4  3  2  1 0 1 2 3 4 5 6 7 8 You can use the following properties to solve absolute-value equations.

  13. means SOLVING ABSOLUTE-VALUE EQUATIONS |a x b |  c a x b  c ora x b   c.

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