1. Objective SWBAT solve absolute value equations.

2. ABSOLUTE VALUE –The distance a number is away from ZERO. Distance is always positive. 012345-2-3-4-5 negative positivezero Remember This??? What is the absolute value of -4? or |-4| |-4| = 4

3. ABSOLUTE VALUE EQUATION– an equation that contains an absolute value expression. Section 6.5 “Solve Absolute Value Equations” |x| = 4 means the distance ‘x’ is from zero. The solution to |x| = 4 is 4 and -4 because they are the only numbers whose distance from 0 is 4. 012-23-4-34

4. Solving an Absolute Value Equation The equation |ax + b| = c where c ≥ 0, is equivalent to the statement: is equivalent to the statement: ax + b = c or ax + b = -c ax + b = c or ax + b = -c

5. Rewrite as two equations. Solve |x – 3| = 8 Rewrite the absolute value equations as two equations. Then solve each equation separately. x – 3 = 8 or x – 3 = -8 x – 3 + 3 = 8 + 3 or x – 3 + 3 = -8 + 3 Addition or Subtraction property of inequality x = 11 or x = -5 Simplify. Solve an Absolute Value Equation Solve an Absolute Value Equation CHECK |x – 3| = 8. |x – 3| = 8. |11 – 3| = 8. |11 – 3| = 8. |-5 – 3| = 8. |-5 – 3| = 8. 8 = 8. 8 = 8. Substitute for x. Simplify.

6. Rewrite as two equations. Solve |r – 7| = 9 Rewrite the absolute value equations as two equations. Then solve each equation separately. Addition or Subtraction property of inequality Simplify. Solve an Absolute Value Equation Solve an Absolute Value Equation CHECK |r – 7| = 9. |r – 7| = 9. |16 – 7| = 9. |16 – 7| = 9. |-2 – 7| = 9. |-2 – 7| = 9. 9 = 9. 9 = 9. Substitute for r. Simplify. r – 7 = 9 r– 7 = 9 or r – 7 = –9 r = 16 or r = –2

7. Solve 3|2x – 7| - 5 = 4 Solve an Absolute Value Equation Solve an Absolute Value Equation First, rewrite the equation in the form ax + b = c. 3 2x – 7 – 5 = 4 3 2x – 7 = 9 2x – 7 = 3 2x – 7 = 3 Write original equation. Add 5 to each side. Divide each side by 3. Next, solve the absolute value equation. 2x – 7 = 3 2x – 7 = 3 or 2x – 7 = –3 2x = 10 or 2x = 4 x = 5 or x = 2 Write absolute value equation. Rewrite as two equations. Add 7 to each side. Divide each side by 2.

8. Solve 4|t + 9| - 5 = 19 Solve an Absolute Value Equation Solve an Absolute Value Equation First, rewrite the equation in the form ax + b = c. Next, solve the absolute value equation. 4 t + 9 – 5 = 19 4 t + 9 = 24 t + 9 = 6 t + 9 = 6 Write original equation. Add 5 to each side. Divide each side by 4. t + 9 = 6 t + 9 = 6 or t + 9 = –6 t = –3 or t = –15 Write absolute value equation. Rewrite as two equations. Addition & subtraction to each side

9. Solving an Absolute Value Equation, if possible… The equation |ax + b| = c where c ≥ 0, is equivalent to the statement: is equivalent to the statement: ax + b = c or ax + b = -c ax + b = c or ax + b = -c The equation |ax + b| = c where c < 0, is NO SOLUTION. is NO SOLUTION.

10. Solve |3x + 5| + 6 = -2, if possible. Solve an Absolute Value Equation Solve an Absolute Value Equation First, rewrite the equation in the form ax + b = c. 3x + 5 + 6 = -2 3x + 5 + 6 = -2 3x + 5 = -8 3x + 5 = -8 Write original equation. Subtract 6 from each side. 3x + 5 + 6 = -2 3x + 5 + 6 = -2 - 6 The equation |ax + b| = c where c < 0, is NO SOLUTION. is NO SOLUTION.

11. Solve -9|4p + 2| - 8 = -35, if possible Solve an Absolute Value Equation Solve an Absolute Value Equation First, rewrite the equation in the form ax + b = c. Write original equation. Add 8 to each side. Divide each side by -9. 4p + 2 = 3 or 4p + 2 = –3 p = ¼ or p = -1¼ Write absolute value equation. Rewrite as two equations. Addition & subtraction to each side -9|4p + 2| – 8 = -35 -9|4p + 2| = -27 +8 |4p + 2| = 3

12. Solve |x - 1| + 5 = 2, if possible. Solve an Absolute Value Equation Solve an Absolute Value Equation First, rewrite the equation in the form ax + b = c. x - 1 + 5 = 2 x - 1 + 5 = 2 x – 1 = -3 x – 1 = -3 Write original equation. Subtract 5 from each side. x - 1 + 5 = 2 x - 1 + 5 = 2 - 5 The equation |ax + b| = c where c < 0, is NO SOLUTION. is NO SOLUTION.

13. Absolute Deviation The absolute deviation of a number x from a given value is the absolute value of the difference of x and the given value. Absolute deviation = |x – given value| Find the values of x that satisfy the definition of absolute deviation for the given value of x: Given Value: 5 Absolute deviation: 8 Absolute deviation = |x – given value| 8 = |x – 5| 8 = |x – 5| 8 = x – 5 -8 = x – 5 13 = x -3 = x

14. Homework –Text p. 393, #4-30 evens, 34-38 even