1. Do Now: Evaluate: Aim: How Do We Solve Absolute Value Equations? 1. 2.
Solve for x : 3. 4.

2. Whatever the value of x might be, taking the absolute value of x makes it positive. Since x might have been positive and might have been negative, you have to acknowledge this fact when you take the absolute-value bars off, and you do this by splitting the equation into two cases.

3. Since the absolute of any number only come out positive, then the absolute value of x + 2 can only be positive.
If the value of x + 2 was positive to start with, then you can bring that value out of the absolute-value bars without changing its sign, giving you x = 4.
But x + 2 might also have been negative, in which case you would have to change the sign on (x + 2) for the absolute value to come out positive, so you also have –(x + 2) = 6 or –x – 2= 6, which solves as x = –8.

4. 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.

5. 6|5x + 2| = 312 6|5x + 2| = 312 |5x + 2| = 52 5x + 2 = 52 5x + 2 = -52
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 = x = -54
x = 10 or x = -10.8
Check: 6|5x + 2| = |5x + 2| = 312
6|5(10)+2| = |5(-10.8)+ 2| = 312
6|52| = |-52| = 312
312 = = 312

6. 3|x + 2| -7 = 14 3|x + 2| -7 = 14 3|x + 2| = 21 |x + 2| = 7
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 = x + 2 = -7
x = or x = -9
Check: 3|x + 2| - 7 = |x + 2| -7 = 14
3|5 + 2| - 7 = |-9+ 2| -7 = |7| - 7 = |-7| -7 = 14
= = 14
14 = = 14

7. Try the following on your own
1. Solve | 2x – 3 | – 4 = 3
X = 5, -2
2. Solve
No Solution
3. Solve
X = 5, -6