Question

How do you solve `(2x-1)/(3x-2) <=1`?

Answer 1

Solve `(2x-1)/(3x-2)<=1`

Multiply both sides by `3x-2`

`(cancel (3x-2)xx2x-1)/cancel(3x-2)<=1xx(3x-2)` =

`2x-1<=3x-2`

Subtract `3x` from both sides.

`2x-1-3x<=-2`

Add `1` to both sides.

`2x-3x<=-2+1` =

`-x<=-1`

Multiply by `-1`

`x>=1`

Answer 2

`(2x-1)/(3x-2) <= 1`

Case 1:
If `3x-2<0`
which implies `color(red)(x<2/3)`
then
`2x-1 >= 3x-2`
( since multiplying by a negative reverses the inequality)
and
`color(red)(x<= 1)`
Combining the Case 1 restrictions on `x`
`color(red)(x<2/3)` and `color(red)(x<=1)`
gives
`color(red)(x<2/3)`

Case2
If `3x-2>0`
which implies `color(blue)(x>2/3)`
then
`2x-1<=3x-2`
`color(blue)(x>=1)`
Combining the Case 2 restrictions on `x`
`color(blue)(x>2/3)` and `color(blue)(x>=1)`
gives
`color(blue)(x>=1)`

Solution
`x<2/3` or `x>=1`