1. 1.7: Solve Absolute Value Equations and Inequalities
Objectives:
To solve and graph absolute value equations and inequalities

2. Exercise 1
Kenny digs a hole in his backyard while his parents are still at work. He neatly piles the dirt from the hole on the concrete patio. When Kenny’s parents return home, they demand that Kenny explain why he destroyed their well-manicured backyard oasis. Kenny says that the hole and the dirt from the whole demonstrate absolute value. What is Kenny talking about?

3. Absolute Value
The absolute value of a number is its distance from zero on a real number line.
The absolute value is always positive.

4. Absolute Value Equations

5. Solving Absolute Value Equations
To solve |ax + b| = c,
Write TWO equations:
ax + b = c
ax + b = -c
Solve each equation.
Check each solution in the original equation.

6. Exercise 2
Solve each absolute value equation.
|x| = 15
|2x – 9| = 15.

7. Exercise 3
Solve.
|4x + 12| = 28
|4x + 10| = 6x

8. Extraneous Solutions
In the course of solving an absolute value equation, one of the solutions may not actually satisfy the original equation. This an extraneous solution. Get rid of it; it’s no good!

9. Try in your notebooks 
Page , 9-15, 21-24, 34-36

10. Absolute Value Inequalities
Follow me here:
|x| = 5 means the distance from zero equals 5.
|x| ≤ 5 means the distance from zero is less than or equal to five.
-5 ≤ x ≤ 5

11. Absolute Value Inequalities
Follow me here:
|x| = 5 means the distance from zero equals 5.
|x| ≥ 5 means the distance from zero is greater than or equal to five.
x ≤ -5 or x ≥ 5

12. Absolute Value Inequalities
If the general methods from the previous slides are incomprehensible, you could just memorize these.

13. Exercise 4 Solve the inequality. Then graph the solution.

14. Exercise 5 Solve the inequality. Then graph the solution. |x + 4| ≥ 6

15. Exercise 6
A professional baseball should weigh ounces, with a tolerance of ounces. (Tolerance is the maximum deviation from the ideal measurement.) Write and solve an absolute value inequality that describes the acceptable weights of a baseball.

16. Solution First write a verbal model of the equation:
│actual weight- ideal weight │≤ tolerance
Convert verbal model to an equation and solve:
│w │≤ 0.125
-0.125≤ w-5.125≤ 0.125
5 ≤ w ≤ 5.25

17. Assignment
P. 55: 40,43-54,74, 76a-b