Question

If `abs(x-2)<6`, then what are the bounds of `x`? i.e. find `a` and `b` such that `a<x<b`.

Answer 1

`a=-4` and `b=8`

Explanation:

`abs(x-2) < 6` if and only if `-6 < x-2 < 6`

if and only if `-6+2 < x-2+2 < 6+2`

if and only if `-4 < x < 8`.

A graph often helps to visualise inequalities:
graph{(y-|x-2|)(y-6)=0 [-6.5, 13.5, -1.4, 8.6]}

`abs(x-2) < 6` is the region below the line `y=6` which you can see is the interval `-4 < x < 8`.

Answer 2

`abs (x-2) < 6" "<=>" ""-"6< x-2<6" "<=>" ""-"4< x<8`.

`a="-"4, b=8.`

Explanation:

When an absolute value inequality has an upper limit (like `abs (x-2) < 6` does), what it means is that the magnitude of `x-2` needs to be under 6; in other words, `x-2` can be positive or negative, but it must be within 6 units of 0, either way.

This means `x-2` has a lower limit of `-6`, but also an upper limit of `6`. We can then rewrite `abs(x-2)<6` as

`"-"6 < x-2 < 6`.

The final step is to isolate `x`. We do this by adding 2 to all "sides" of the inequality, to get

`"-"6+color(red)2 < x - 2 + color(red)2 < 6 + color(red)2`

or

`"-"4< x < 8`.

Thus, the lower limit of `x` is `a="-"4`, and the upper limit of `x` is `b=8`.