Question

How do you solve and write the following in interval notation: `(x-5)/3 - (4x+3)/6> -3`?

Answer 1

`x in (-oo, 5/2)`

Explanation:

The first thing to do here is get rid of the denominators by multiplying the first fraction by `1 = 2/2` and `-3`, which can be thought of as a fraction with the denominator equal to `1`, by `12 = 6/6`.

The inequality becomes

`(x-5)/3 * 2/2 - (4x+3)/6 > -3 * 6/6`

`(2(x-5))/6 - (4x+3)/6 > -18/6`

This will be equivalent to

`2(x-5) - 4x - 3 > -18`

Expand the parentheses and group like terms to find

`2x - 10 - 4x - 3 > -18`

`-2x - 13 > -18`

`-2x > -5`

Now, you must divide both sides of the inequality by `-2` to isolate `x`. Do not forget that multiplying or dividing by a negative number changes the sign of the inequality!

In this case, you have

`> -> <`

and

`(color(red)(cancel(color(black)(-2)))x)/(color(red)(cancel(color(black)(-2)))) < (-5)/(-2)`

`x < 5/2`

This tells you that any value of `x in RR` that is smaller than `5/2` will be a solution to the given inequality. Mind you, `5/2` is not part of the solution interval.

In interval notation, this can be written as

`color(green)(|bar(ul(color(white)(a/a)color(black)(x in (-oo, 5/2))color(white)(a/a)|)))`

graph{x<5/2 [-10, 10, -5, 5]}