Question

How do you solve `5< 3- 2x/3`?

Answer 1

`x < -12 or x < -3`
`color(white)("XXXX")`depending upon interpretation of `2x/3`

Explanation:

Note I have interpreted `2x/3` as a mixed fraction; if it were intended to be a multiplication, see Solution 2 below.

Solution 1 (`2x/3` as a mixed fraction)
If `5 < 3 - 2x/3` `color(white)("XXXX")` `rarr (5 < 3 - (6+x)/3)`
We can clear the fraction by multiplying everything by 3 (without effecting the validity of the inequality orientation)
`color(white)("XXXX")` `15 < 9-6 -x`

Subtract 3 from both sides (again, no effect on the validity of the inequality orientation)
`color(white)("XXXX")` `12 < -x`

Multiply both sides by `(-1)` (This will reverse the inequality orientation. Note that as an alternative we could have added `(x-12)` to both sides without effecting the orientation and achieve the same result)
`color(white)("XXXX")` `x < -12`

Solution 2 (`2x/3` as implied multiplication)
If `5 < 3 - 2x/3` `color(white)("XXXX")` `rarr (5 < 3 - 2/3x)`

Subtract 3 from both sides
`color(white)("XXXX")` `2 < -2/3x`

Multiply both sides by `(-3/2)` (forcing a reversal of the inequality orientation
`color(white)("XXXX")` `-3 > x`
or
`color(white)("XXXX")`x < -3#