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

Apr 17, 2018

$x = 1$

#### Explanation:

Assuming you want to solve for x, find the common denominator for the left side, $\left(x + 1\right) \left(x - 4\right)$ and combine them into a single rational equation. Since $\frac{4}{x + 1}$ and $\frac{3}{x - 4}$ combine, it turns into $\frac{7 x - 13}{\left(x + 1\right) \left(x - 4\right)}$ by multiplying the numerator $4 \cdot \left(x - 4\right)$ and $3 \cdot \left(x + 1\right)$, adding them and combining like terms . You want to get rid of any denominators in the problem to make it simpler, so multiply that common denominator to both sides of the equation,
$\frac{7 x - 13}{\left(x + 1\right) \left(x - 4\right)} \cdot \left(x + 1\right) \left(x + 4\right)$ and same with the right side. On the left the common denominator gets cancelled so you're left with $7 x - 13$. On the right the $\left(x + 1\right)$ gets cancelled and you have $2 \left(x - 4\right)$. Multiply that and overall you have $7 x - 13 = 2 x - 8$ left. then bring the $x ' s$ on one side and the rest on the other side. $7 x - 2 x = - 8 + 13$ , add and finally you have, $5 x = 5$, $x = 1$. To check you can plug $x$ in the original equation.

Apr 17, 2018

$x = 1$

#### Explanation:

$\frac{4}{x + 1} + \frac{3}{x - 4} = \frac{2}{x + 1}$

Our strategy will be to eliminate quotients.

This equation will look a lot less scary if we multiply both sides by $x + 1$.

$4 + \frac{3 \left(x + 1\right)}{x - 4} = 2$

Now subtract 4 from both sides of the equation.

$\frac{3 \left(x + 1\right)}{x - 4} = - 2$

Now multiply both side s of the equation by $x - 4$.

$3 \left(x + 1\right) = - 2 \left(x - 4\right)$

Hey! No more quotients! Now apply the distributive property.

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

Add $2 x$ to both sides of this equation.

$5 x + 3 = 8$

Subtract 3 from both sides of this equation.

$5 x = 5$

Divide both sides of this equation by 5.

$x = 1$

$\frac{4}{1 + 1} + \frac{3}{1 - 4}$ =?= $\frac{2}{1 + 1}$
$\frac{4}{2} + \frac{3}{-} 3$ =?= $\frac{2}{2}$
$2 - 1$ =?= $1$
$1 = 1$