Question

The given equation is either linear or equivalent to a linear equation. Solve the equation.

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

Short answer

The solution of the given equation is $x=\frac{13}{3}$.

Step-by-step solution

Step 1. Definition of equation.

Equation is a mathematical statement in which two expressions are equal to each other.

Step 2. Given Information.

The given equation is:

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

Step 3. Solve the given equation.

The given equation can be written as:

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

The LCM of$x+1,2,3\left(x+1\right)$ is $6\left(x+1\right)$, so multiply both sides by$6\left(x+1\right)$ to simplify the given equation.

role="math" localid="1643312273816" $$\begin{gathered} 6\left(x+1\right)\left(\frac{3}{x+1}-\frac{1}{2}\right)=6\left(x+1\right)\left(\frac{1}{3\left(x+1\right)}\right) \\ 6\left(x+1\right)\left(\frac{3}{x+1}\right)+6\left(x+1\right)\left(-\frac{1}{2}\right)=2\left(1\right) \\ 6\left(3\right)-3\left(x+1\right)=2 \\ 18-3x-3=2 \\ 15-3x=2 \end{gathered}$$

Subtract$15$ from both sides.

role="math" localid="1643312327120" $\begin{array}{c}15-3x-15=2-15\\ -3x=-13\end{array}$

Divide both sides by $-3$.

$\begin{array}{c}\frac{-3x}{-3}=\frac{-13}{-3}\\ x=\frac{13}{3}\end{array}$

Thus, the solution of the given equation is $x=\frac{13}{3}$.