Question

How do you solve the inequality `|1-x/3|>2/3`?

Answer 1

Consider each case for the inequality being true to find the solution set `(-oo,1)uu(5,oo)`

Explanation:

We can solve this if we recognize what the absolute value symbol is actually saying. It is equivalent to saying

`1-x/3 > 2/3`
or
`1-x/3 < -2/3`

Then all we need to do is find what conditions cause each to be true.

For the first:
`1-x/3 > 2/3`
`=> 1-2/3 > x/3`
`=> 1/3 > x/3`
`=> 1 > x`

For the second:
`1-x/3 < -2/3`
`=> 1+2/3 < x/3`
`=> 5/3 < x/3`
`=> 5 < x`

So one of the conditions, and thus the original inequality, will be true so long as either `x < 1` or `x > 5`. Therefore we have the solution set
`(-oo,1)uu(5,oo)`