Question

How do you solve `(x+68)/(x+8)>=5`?

Answer 1

`x in (-8, 7]`

Explanation:

Given:

`(x+68)/(x+8) >= 5`

Subtract `(x+68)/(x+8)` from both sides to get:

`0 >= 5-(x+68)/(x+8) = (5(x+8)-(x+68))/(x+8) = (4x-28)/(x+8) = (4(x-7))/(x+8)`

Note that the right hand side is a function which is continuous and non-zero except at `x=-8` and `x=7`. The function changes sign at each of these two points.

When `x=-8` the denominator is `0` so the right hand side is undefined. So `-8` is not part of the solution set.

When `x=7` the numerator is `0` and the inequality is satisfied. So `7` is part of the solution set.

When `x < -8` or `x > 7` then the signs of the numerator and denominator are the same, so the quotient is positive and the inequality is not satisfied.

When `x in (-8, 7)` the numerator is negative, the denominator is positive and the quotient is negative. So the inequality is satisfied.

So the solution set is:

`x in (-8, 7]`

Answer 2

The solution is `x in (-8,7]`

Explanation:

We cannot do crossing over.

So, we simplify the inequality

`(x+68)/(x+8)>=5`

`(x+68)/(x+8)-5>=0`

`((x+68)-5(x+8))/(x+8)>=0`

`(x+68-5x-40)/(x+8)>=0`

`(28-4x)/(x+8)>=0`

`(4(7-x))/(x+8)>=0`

Let `f(x)=(4(7-x))/(x+8)`

We can build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-8` `color(white)(aaaaaaaa)` `7` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x+8` `color(white)(aaaa)` `-` `color(white)(aaa)` `||` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `7-x` `color(white)(aaaa)` `+` `color(white)(aaa)` `||` `color(white)(aaaa)` `+` `color(white)(aaaa)` `-`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaa)` `-` `color(white)(aaa)` `||` `color(white)(aaaa)` `+` `color(white)(aaaa)` `-`

Therefore,

`f(x)>=0` when `x in (-8,7]`