Are these two piecewise functions equivalent or different?

Question

I got this solution to the inequality $|x - 5| + |x - 4| \ge 3:$

$$|x - 5| + |x - 4| - 3 = \begin{cases} 2x - 12& \text{if }x > 5\\ -2 & \text{if }4 \le x \le 5\\ -2x + 6& \text{if }x < 4\\ \end{cases}. $$

My professor said that it should be $2x - 12 \text{ if } x \ge 5.$

Whose answer is correct? Or are both technically correct?

Answer 1

An image says more than a thousand words :-)

($h$ is the one in bold)