# How do you solve (2x)/(x-4)=8/(x-4)+3?

May 14, 2017

No solution!

#### Explanation:

First, note that 4 cannot be a solution (division by zero)

Then, multiply both sides by $\left(x - 4\right)$, you get

$2 x = 8 + 3 \cdot \left(x - 4\right)$
$\implies 3 x - 2 x = 3 \cdot 4 - 8$
$\implies x = 4$ which is impossible!
So there is no solution

May 14, 2017

The equation is unsolvable.

#### Explanation:

We have:

$\frac{2 x}{x - 4} = \frac{8}{x - 4} + 3$

Multiply all terms by $x - 4$.

$2 x = 8 + 3 \left(x - 4\right)$

Expand the brackets.

$2 x = 8 + 3 x - 12 = 3 x - 4$

Add $4 - 2 x$ to both sides.

$4 = x$ or $x = 4$

Unfortunately this leads to a problem that $x = 4$ is a singularity (mathematicians don't like infinities).

Reorganise the original equation by subtracting $\frac{8}{x - 4}$ from both sides.

$\frac{2 x - 8}{x - 4} = 3$

This gives:

$\frac{2 \left(x - 4\right)}{x - 4} = 3$

This gives $2 = 3$ the equation makes no sense unless $x = 4$ in which case both sides are infinite..