Question

How do you solve `\frac { x + 3} { x - 7} \leq 0`?

Answer 1

Answer is `x` lies in the interval `-3 <= x < 7` or `[-3,7)`.

Explanation:

First let us work for equality. As `(x+3)/(x-7)=0` we should have `x+3=0` i.e. `x=-3`.

Now for inequality `(x+3)/(x-7)<0`, this means `(x+3)/(x-7)` is negative, which can be on two conditions

(1) either `x+3>0` (i.e. it is positive) and `x-7<0` (i.e. it is negative). This means `x> -3` and `x<7` i.e. `x` lies between `-3` and `7`. In other words it lies in the interval `-3 < x < 7`.

(2) or `x+3<0` (i.e. it is negative) and `x-7>0` (i.e. it is positive). This means that `x<-3` and `x>7`. But this cannot be.

Hence including `x=-3`, the answer is `x` lies in the interval `-3 <= x < 7` or `[-3,7)`.