Question

How do solve `(3x+1)/(x+4)<=1` and write the answer as a inequality and interval notation?

Answer 1

The solution is `x in (-4,3/2]` or `-4 < x <= 3/2`

Explanation:

Let's rewrite and simplify the inequality

`(3x+1)/(x+4)<=1`

`(3x+1)/(x+4)-1<=0`

`((3x+1)-(x+4))/(x+4)<=0`

`((2x-3))/(x+4)<=0`

Let `f(x)=((2x-3))/(x+4)`

Now, we can build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaa)` `-4` `color(white)(aaaaaaa)` `3/2` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x+4` `color(white)(aaaaaa)` `-` `color(white)(aa)` `0` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `2x-3` `color(white)(aaaaa)` `-` `color(white)(aa)` ###color(white)(aaaaa)#`-` `color(white)(aa)` `0` `color(white)(aa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaa)` `+` `color(white)(aa)` `||` `color(white)(aaaa)` `-` `color(white)(aa)` `0` `color(white)(aa)` `+`

Therefore,

`f(x)<=0` when `x in (-4,3/2]` or `-4 < x <= 3/2`

graph{(3x+1)/(x+4)-1 [-27.09, 18.51, -12.22, 10.59]}