Simplify: |z-6|-|z-5| if z<5

Answer 1

Simplify | z - 6 | - | z - 5 | if z < 5.

Well, the definition of absolute value is:

| a | = a, if a >= 0, and

| a | = -a, if a < 0.

And if z < 5, then (z - 6) and (z - 5) are both < 0, so

| z - 6 | - | z - 5 | = -(z - 6) - (- (z - 5))

= -z + 6 + (z - 5)

= 1

Answer 2

I would simply try a few different values of Z:

Suppose that Z = 8. We would then have |8-6| - |8-5| = 2 - 3 = -1

Next, try Z = 4. This gives |4-6| - |4-5| = 2 - 1 = +1

I think you'll find that, when Z < 5, the first term will alwasy have the larger absolute value, and so the expression value will be +1