Question

Write each of the following expressions without using absolute value. |z−7|−|z−9|, if z<7

Answer 1

Here's my thinking:  If z<7, then z-7 is negative, and |z-7| is positive.  So, with the understanding that z-7 is positive, the first part of this expression becomes:

7-z for z<7

Similarly:  if z<7, then z-9 is negative and |z-9| is positive.  Thus, -|z-9| is equivalent to (9-z).


|z−7|−|z−9|, if z<7 becomes (7-z) - (9-z) for z<7.  Simplifying this, we get:

7 - z - 9 + z = -2