1. Solving Absolute Value Equations

2. What is Absolute Value?
The absolute value of a number is the number of units it is from zero on the number line.
5 and -5 have the same absolute value.
The symbol |x| represents the absolute value of the number x.

3. Absolute Value can also be defined as :
|-8| = 8
|4| = 4
You try:
|15| = ?
|-23| = ?
Absolute Value can also be defined as :
if a >0, then |a| = a
if a < 0, then |a| = -a

4. We can 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. Equations may also contain absolute value expressions
When solving an equation, isolate the absolute value expression first.
Rewrite the equation as two separate equations.
Consider the equation | x | = 3. The equation has two solutions since x can equal 3 or -3.
Solve each equation.
Always check your solutions.
Example: Solve |x + 8| = 3
x + 8 = 3 and x + 8 = -3
x = x = -11
Check: |x + 8| = 3
|-5 + 8| = 3 | | = 3
|3| = |-3| = 3
3 = = 3

6. Now Try These Solve |y + 4| - 3 = 0
|y + 4| = You must first isolate the variable by adding to both sides.
Write the two separate equations.
y + 4 = 3 & y + 4 = -3
y = y = -7
Check: |y + 4| - 3 = 0
|-1 + 4| -3 = 0 |-7 + 4| - 3 = 0
|-3| - 3 = |-3| - 3 = 0
3 - 3 = = 0
0 = = 0

7. Absolute value is never negative.
|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!

8. Solve: 3|x - 5| = 12
|x - 5| = 4
x - 5 = 4 and x - 5 = x = x = 1
Check: 3|x - 5| = 12
3|9 - 5| = |1 - 5| = 12
3|4| = |-4| = 12
3(4) = (4) = 12
12 = = 12

9. Solve: |8 + 5a| = 14 - a
8 + 5a = 14 - a and a = -(14 – a)
Set up your 2 equations, but make sure to negate the entire right side of the second equation.
8 + 5a = 14 - a and a = a
6a = a = a = a = -5.5
Check: |8 + 5a| = 14 - a
|8 + 5(1)| = |8 + 5(-5.5) = 14 - (-5.5)
|13| = |-19.5| = = = 19.5