Question

1. For what values of x does the function y=2x+13 take on a positive value? a negative value?

Answer 1

$2x+13=0\\ 2x=-13\\ x=-6.5$

The function is linear where $a\ \textgreater \ 0$, so it's negative for $x\ \textless \ -6.5$ and positive for $x\ \textgreater \ -6.5$

............................

$D:x\in\mathbb{R}\setminus\{-1,1,0\}\\\\ \dfrac{a+3}{a^2-1} - \dfrac{1}{a^2+a}=\\ \dfrac{a+3}{(a-1)(a+1)} - \dfrac{1}{a(a+1)}=\\ \dfrac{a(a+3)}{a(a-1)(a+1)} - \dfrac{a-1}{a(a-1)(a+1)}=\\ \dfrac{a^2+3a-a+1}{a(a-1)(a+1)}=\\ \dfrac{a^2+2a+1}{a(a-1)(a+1)}=\\ \dfrac{(a+1)^2}{a(a-1)(a+1)}=\\ \dfrac{a+1}{a(a-1)}$

for $a\in(-\infty,-1)$ it's negative
for $a\in(-1,0)$ it's positive
for $a\in(0,1)$ it's negative
for $a\in(1,\infty)$ it's positive