# How do you solve 5/(2x+7) = -9/(4x+14)?

Sep 27, 2015

This equality has no solutions.

#### Explanation:

An identity holds if and only if the identity between the inverses hold (as long as you don't have something like $0 = 0$ of course). So, in you case, you have that

$\frac{5}{2 x + 7} = - \frac{9}{4 x + 14} \setminus \iff \frac{2 x + 7}{5} = \frac{4 x + 14}{9}$

The only thing we have to take care about is to make sure that the original denominators aren't zero, i.e. $2 x + 7 \setminus \ne 0$ and $4 x + 14 \setminus \ne 0$, both satisfied by $x \setminus \ne - \frac{7}{2}$.

Now we can go on searching for the solutions:

$\frac{2 x + 7}{5} = \frac{4 x + 14}{9} \setminus \iff 9 \left(2 x + 7\right) = 5 \left(4 x + 14\right)$

by cross-multiplication, and expanding we get

$18 x + 63 = 20 x + 70$. Isolating the $x$-terms and the costants, we finally get

$- 2 x = 7$, and finally solve for $x = - \frac{7}{2}$

This is the value we couldn't consider, so the equality has no solutions.