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

Answer 2

Using the definition of the absolute value function, it is found that the expression for z < 7 is:

$|z - 7| - |z - 9| = -2$

The definition of the absolute value function is:

$|f(x)| = f(x), x \geq 0$

$|f(x)| = -f(x), x < 0$

Then, for |z - 7|.

$z - 7 \geq 0$

$z \geq 7$

Then:

$|z - 7| = z - 7, z \geq 7$

$|z - 7| = -z + 7, z < 7$

For |z - 9|.

$z - 9 \geq 0$

$z \geq 9$

Then:

$|z - 9| = z - 9, z \geq 9$

$|z - 9| = -z + 9, z < 9$

Working with z < 7, we have that:

$|z - 7| = -z + 7$

$|z - 9| = -z + 9$

Then:

$|z - 7| - |z - 9| = -z + 7 - (-z + 9) = -z + 7 + z - 9 = -2$

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