An equation with absolute values in it will have two answers. We'll have the happy-go-lucky positive answer, and the sourpuss negative answer.
The first answer comes from keeping the inside of the absolute value the same, and solving.
3x + 9 = 15
3x = 6
x = 2
The second answer comes from finding the negative of what's inside the bars.
-(3x + 9) = 15
Let's multiply out the negative by -1, instead of distributing it.
3x + 9 = -15
3x = -24
x = -8
Our final answer is x = -8 or 2.
Example 2
Solve
One equation enters, two answers leave. The first just walks right out of its cage.
How about we divide the 2 out of our fraction, and then multiply each side by 9 after that? That sounds better than the alternatives, one of which involves 6 rubber chickens, a copy of Battlefield Earth, and the longest Friday night of our life.
x = 81
The second answer needs to put on a clever disguise to get out. A mop, some rags lying around, and a thick accent come together for a getaway that would make Scooby Doo and Shaggy proud.
x = -90
Our answer is x = 81 or -90. Shmoopy-doopy doo!
Example 3
Solve |11 – 3x| = 13.
Can we rock this question to find the answers? Absolutely.
11 – 3x = 13
-3x = 2
One down, one to go.
-(11 – 3x) = 13
This time, we will distribute the -1, so that we can get rid of our negative coefficient on x.
or x = 8.