Question

Solve the absolute value of inequalities and express in interval notation for `abs(3(x+2)-7x) <=6`?

Answer 1

Simplify the expression within the absolute value then consider the two cases where that expression has a negative and a non-negative value.

`abs(3(x+2)-7x) <=6`

`abs(6-4x)<=6`

Case 1
If `6-4x<0`
then `abs(6-4x) = 4x-6`
and `x>3/2`

`4x-6<=6`
`4x<=12`
`x<=3`

So in this case `3/2<x<=3`

Case 2
If `6-4x>=0`
then `abs(6-4x) = 6-4x`
and `x<=3/2`

`6-4x <=6`
`-4x<=0`
`x>=0`

In this case `0<=x<=3/2`

Combining the cases
Valid solutions are values of `x` for which
either `3/2<x<=3`
or `0<=x<=3/2`

which combines as
`0<=x<=3`