| x - 2 | = 4x + 4 is our given equation. In an absolute value equation, we solve the original expression as our first equation. Our second one is that we multiply the right side by -1 (using the definition of absolute value)
Case 1: original equation
|x - 2| = 4x + 4 <-----original
x - 2 = 4x + 4 <-----take the positive one
-2 = 3x + 4 <-----subtract x on both sides
-6 = 3x <-----subtract 4 on both sides
-2 = x <-----divide both sides by -3
Case 2: Opposite equation
| x - 2 | = 4x + 4 <------original equation
x - 2 = -(4x + 4) <------take the negative of the right side
x - 2 = -4x -4 <----multiply the right side by -1
-2 = -5x - 4 <------subtract x on both sides
2 = -5x <-------add 4 on both sides
x = -2/5 <--------divide both sides by -5
Now we have two solutions. We need to check for extraneous solutions because of all the manipulations.
Check:
| x - 2 | = 4x + 4 <------- original
| -2 -2 | = 4(-2) + 4 <--------use x = -2
| -4 | = -8 + 4
4 = -4 Not a solution
| x - 2 | = 4x + 4 <------- original
| -2/5 - 2 | = 4(-2/5) + 4 <----use x = -2/5
|-2/5 - 2 | = -8/5 + 4
|-2/5 - 10/5 | = -8/5 + 20/5
|-12/5 | = 12/5
12/5 = 12/5 Solution
Therefore x = -2/5