Question

How do you solve `Solve: 1/2x - 3 <=-1/1.5x - .5 <=1/2x + 2`?

Answer 1

Given `1/2x-3 <= -1/1.5x =.5 <= 1/2x+2`

Simplify by multiplying each expression by `6` to clear the fractions (Remember that you can multiply by any value greater than zero without effecting the orientation of the inequalities).
`3x-9 <= -4x-3 <= 3x+12`

Break this up into two compound inequalities:
`"[1] "3x-9<=-4x-3`
and
`"[2] "-4x-3<=3x+12`

Evaluate Inequality [1]
`3x-9<=-4x-3`
`color(white)("MMMMMM")`Add `(4x+9)` to both sides:
`7x <= 6`
`color(white)("MMMMMM")`Divide by `7` (which does not change the inequality orientation)
`x<=6/7`

Evaluate Inequality [2]
`-4x-3 <= 3x+12`
`color(white)("MMMMMM")`Add `(4x-12)` to both sides
`-15<= 7x`
`color(white)("MMMMMM")`Divide both sides by 7
`(-15/7)<=x`

Combine the Compound Inequalities (with and)
`x<=6/7` and `x>= (-15/7)`
`color(white)("MMMMMM")`Re-write as a single compound statement
`(-15/7) <= x <= 6/7`