We simply drop the absolute value signs and write whatever is inside
This is because |x-4| is really composed of two equations and it depends on what x is
If x < 4, then we use -(x-4) or -x+4
If x >= 4, then we use x-4
This is from the idea that |x| breaks down into this piecewise function
[tex]|x| = \begin{cases}x \text{ if } x \ge 0 \\ -x \text{ if } x < 0\end{cases}[/tex]
Replace every x with x-4
[tex]|x-4| = \begin{cases}x-4 \text{ if } x-4 \ge 0 \\ -(x-4) \text{ if } x-4 < 0\end{cases}[/tex]
Then for each inequality, you would solve for x to get the following
[tex]|x-4| = \begin{cases}x-4 \text{ if } x \ge 4 \\ -(x-4) \text{ if } x < 4\end{cases}[/tex]
which is the same as
[tex]|x-4| = \begin{cases}x-4 \text{ if } x \ge 4 \\ -x+4 \text{ if } x < 4\end{cases}[/tex]
Answer: x-4
I used this during a quiz in RSM so its right:)
Thank you to jimthompson5910
