Question

I need help solving this. |x−4| if x>4. You have to write the expression without using the absolute value sign.

Answer 1

Answer:   x-4

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

$|x| = \begin{cases}x \text{ if } x \ge 0 \\ -x \text{ if } x < 0\end{cases}$

Replace every x with x-4

$|x-4| = \begin{cases}x-4 \text{ if } x-4 \ge 0 \\ -(x-4) \text{ if } x-4 < 0\end{cases}$

Then for each inequality, you would solve for x to get the following

$|x-4| = \begin{cases}x-4 \text{ if } x \ge 4 \\ -(x-4) \text{ if } x < 4\end{cases}$

which is the same as

$|x-4| = \begin{cases}x-4 \text{ if } x \ge 4 \\ -x+4 \text{ if } x < 4\end{cases}$