Question

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

Answer 1

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

`5/{2x+7} = - 9/{4x+14} \iff {2x+7} /5 = {4x+14}/9`

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

Now we can go on searching for the solutions:

`{2x+7} /5 = {4x+14}/9 \iff 9(2x+7) = 5(4x+14)`

by cross-multiplication, and expanding we get

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

`-2x=7`, and finally solve for `x=-7/2`

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