Question

the set of all values of x for which x-3/x+5>0 is​

Answer 1

To get the fraction positive, both the numerator and the denominator may have the same sign at the same time.

Here the denominator can't be zero, we know it.

Case 1 :-  $x-3>0,\quad x+5>0$

This implies  $x\in(3,\ \infty),\quad x\in(-5,\ \infty)$

We take the intersection of these two, so we get  $x\in(3,\ \infty)$

Case 2 :-  $x-3<0,\quad x+5<0$

This implies  $x\in(-\infty,\ 3),\quad x\in(-\infty,\ -5)$

Again we take the intersection of these two, so we get  $x\in(-\infty,\ -5)$

Hence the domain is  $(-\infty,\ -5)\cup(3,\ \infty)$

Or the domain is  $\mathbb{R}-[-5,\ 3]$