Find a number, delta, such that if |x-2|<delta, then |4x-8|<0.1

Question

I don't have a clue how to begin this. My teacher just started off the lesson with this without any explanation so I am at a loss.

A step-by-step explanation would be very much appreciated!!

Answer 1

|4x-8| = 4|x-2|
So, if |4x-8| < 0.1
4|x-2| < 0.1
|x-2| < 0.025

So, x is less than 0.025 away from 2. That is,

1.975 < x < 2.025

Or, using the definition of |n|, either
(x-2) < 0.025 if x-2 >= 0
or
-(x-2) < 0.025 if x-2 < 0

So, if x-2 >= 0, x >= 2 and
x < 2.025

If x-2 < 0, x < 2, and
x-2 > -0.025
x > 1.975

So, 1.975 < x < 2.025

Answer 2

Actually, all that stuff above is irrelevant. The question was to find δ.

|4x-8| < .1
4|x-2| < 0.1
|x-2| < 0.025
so, δ = 0.025