Question

Write each of the following expressions without using absolute value: |z−6|−|z−5|, if z<5

Answer 1

Answer:

The answer is 1.

Step-by-step explanation:

Given the expression:

$|z-6|-|z-5|,\ if\ z<5$

To find:

The expression without absolute value.

Solution:

First of all, let us learn about the absolute value function:

$y = f(x) = |x| =\left \{ {{x\ if\ x>0} \atop {-x\ if\ x<0}} \right.$

i.e. value is x if x is positive

value is -x if x is negative

Here the given expression contains two absolute value functions:

$|z-6|$ and $|z-5|$

Using the definition of absolute value function as per above definition.

$|z-5| =\left \{ {{(z-5)\ if\ z>5} \atop {-(z-5)\ if\ z<5}} \right.$

$|z-6| =\left \{ {{(z-6)\ if\ z>6} \atop {-(z-6)\ if\ z<6}} \right.$

Now, it is given that z < 5 that means z will also be lesser than 6 i.e. z < 6

So, given expression $|z-6|-|z-5|,\ if\ z<5$ will be equivalent to :

$-(z-6) - (-(z-5))\\\Rightarrow -z+6 +z-5 = \bold{1}$

So, the expression is equivalent to 1.