Question

How do you solve `p/(p-16)+2/(p-6)<=0` using a sign chart?

Answer 1

The solution is `p in [-4,6) uu [8, 16)`

Explanation:

Let 's simplify the `LHS` of the inequality

`p/(p-16)+2/(p-6)<=0`

`(p(p-6)+2(p-16))/((p-16)(p-6))<=0`

`(p^2-6p+2p-32)/((p-16)(p-6))<=0`

`(p^2-4p-32)/((p-16)(p-6))<=0`

`((p+4)(p-8))/((p-16)(p-6))<=0`

Let `f(p)=((p+4)(p-8))/((p-16)(p-6))`

We can build the sign chart

`color(white)(aaaa)` `p` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-4` `color(white)(aaaaaaa)` `6` `color(white)(aaaaaa)` `8` `color(white)(aaaaaaa)` `16` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `p+4` `color(white)(aaaa)` `-` `color(white)(aaaaaa)` `+` `color(white)(aaa)` `||` `color(white)(aaa)` `+` `color(white)(aaa)` `+` `color(white)(aaa)` `||` `color(white)(aa)` `+`

`color(white)(aaaa)` `p-6` `color(white)(aaaa)` `-` `color(white)(aaaaaa)` `-` `color(white)(aaa)` `||` `color(white)(aaa)` `+` `color(white)(aaa)` `+` `color(white)(aaa)` `||` `color(white)(aa)` `+`

`color(white)(aaaa)` `p-8` `color(white)(aaaa)` `-` `color(white)(aaaaaa)` `-` `color(white)(aaa)` `||` `color(white)(aaa)` `-` `color(white)(aaa)` `+` `color(white)(aaa)` `||` `color(white)(aa)` `+`

`color(white)(aaaa)` `p-16` `color(white)(aaa)` `-` `color(white)(aaaaaa)` `-` `color(white)(aaa)` `||` `color(white)(aaa)` `-` `color(white)(aaa)` `-` `color(white)(aaa)` `||` `color(white)(aa)` `+`

`color(white)(aaaa)` `f(p)` `color(white)(aaaaa)` `+` `color(white)(aaaaaa)` `-` `color(white)(aaa)` `||` `color(white)(aaa)` `+` `color(white)(aaa)` `-` `color(white)(aaa)` `||` `color(white)(aa)` `+`

Therefore,

`f(p)<=0` when `p in [-4,6) uu [8, 16)`