Question

How do you solve and write the following in interval notation: `-4<=( -2x+5)/3<=3`?

Answer 1

`x in [-2, 17/2]`

Explanation:

Your goal here is to isolate `x` in the middle of this compound inequality.

Start by multiplying all sides by `3`

`-4 * 3 <= (-2x + 5)/color(red)(cancel(color(black)(3))) * color(red)(cancel(color(black)(3))) <= 3 * 3`

`color(white)(a)-12 <= color(white)(a)-2x + 5 color(white)(aa)<= 9`

Next, subtract `5` from all sides

`-12 - 5 <= -2x + color(red)(cancel(color(black)(5))) - color(red)(cancel(color(black)(5))) <= 9 - 5`

`color(white)(aaa)-17 <= color(white)(aaa) -2x color(white)(aaaa) <= 4`

Finally, divide all sides by `-2`. Do not forget that when you're multiplying or dividing an inequality by a negative number, the sign of the inequality is flipped!

`(-17)/(-2) color(blue)(>=) color(white)(a)xcolor(white)(a) color(blue)(>=) 4/(-2) ->` the signs are flipped !

will get you

`color(white)(-)17/2 >= color(white)(a)xcolor(white)(a) >= -2`

This tells you that in order to be a solution to this compound inequality, a value of `x` must be greater than or equal to `-2` and smaller than or equal to `17/2`.

In interval notation, this is expressed like this

`color(green)(|bar(ul(color(white)(a/a)color(black)(x in [-2, 17/2])color(white)(a/a)|)))`

You can find this solution interval by breaking up the compound inequality into two simple inequalities

`x <= 17/2" "` and `" "x >= -2`

In interval notation, these solution intervals will get you

`x in (-oo, 17/2] nn [-2, + oo) implies x in [-2, 17/2]`