How do you solve (3x+6)/(x^2-4)=(x+1)/(x-2)?

Aug 24, 2016

$x = 2$

Explanation:

Well there is the long way. Cross multiply
$\left(3 x + 6\right) \left(x - 2\right) = \left({x}^{2} - 4\right) \left(x + 1\right)$
Multiply out solve

Or there is the efficient way. Factorise each term first.
$\frac{3 \left(x + 2\right)}{\left(x + 2\right) \left(x - 2\right)} = \frac{x + 1}{x - 2}$

Cancel and we are left with 3=$x$+1

Aug 24, 2016

This equation has no solution.

Explanation:

$\frac{\frac{3 \left(x + 2\right)}{\left(x + 2\right) \left(x - 2\right)}}{\frac{x + 1}{x - 2}} = 1$

$\frac{3 \left(x + 2\right) \left(x - 2\right)}{\left(x + 2\right) \left(x - 2\right) \left(x + 1\right)} = 1$

Cancelling using the property $\frac{a}{a} = 1$, we are left with:

$\frac{3}{x + 1} = 1$

$3 = 1 \left(x + 1\right)$

$3 = x + 1$

$x = 2$

However, this value of $x$ is extraneous, since it renders the denominator $0$ (which in turn makes the equation undefined).

Hence, this equation has no solution $\left\{\emptyset\right\}$.

Hopefully this helps!