Solve x+1/x+3-1-x/3+2x=2quad(text{iv})

$$\frac { x + 1 } { x + 3 } - \frac { 1 - x } { 3 + 2 x } = 2 \quad ( \text { iv } )$$

Solve for v

$$v=\frac{x\left(3x+7\right)}{2\left(2ix^{2}+9ix+9i\right)}$$

$$x\neq -3\text{ and }x\neq -\frac{3}{2}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x+3\right)\left(2x+3\right), the least common multiple of x+3,3+2x.

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

Use the distributive property to multiply 2x+3 by x+1 and combine like terms.

$$2x^{2}+5x+3-\left(x+3\right)\left(1-x\right)=2iv\left(x+3\right)\left(2x+3\right)$$

Use the distributive property to multiply x+3 by 1-x and combine like terms.

$$2x^{2}+5x+3-\left(-2x-x^{2}+3\right)=2iv\left(x+3\right)\left(2x+3\right)$$

To find the opposite of -2x-x^{2}+3, find the opposite of each term.

$$2x^{2}+5x+3+2x+x^{2}-3=2iv\left(x+3\right)\left(2x+3\right)$$

Combine 5x and 2x to get 7x.

$$2x^{2}+7x+3+x^{2}-3=2iv\left(x+3\right)\left(2x+3\right)$$

Combine 2x^{2} and x^{2} to get 3x^{2}.

$$3x^{2}+7x+3-3=2iv\left(x+3\right)\left(2x+3\right)$$

Subtract 3 from 3 to get 0.

$$3x^{2}+7x=2iv\left(x+3\right)\left(2x+3\right)$$

Use the distributive property to multiply 2iv by x+3.

$$3x^{2}+7x=\left(2ivx+6iv\right)\left(2x+3\right)$$

Use the distributive property to multiply 2ivx+6iv by 2x+3 and combine like terms.

$$3x^{2}+7x=4ix^{2}v+18ixv+18iv$$

Swap sides so that all variable terms are on the left hand side.

$$4ix^{2}v+18ixv+18iv=3x^{2}+7x$$

Combine all terms containing v.

$$\left(4ix^{2}+18ix+18i\right)v=3x^{2}+7x$$

Divide both sides by 18ix+18i+4ix^{2}.

$$\frac{\left(4ix^{2}+18ix+18i\right)v}{4ix^{2}+18ix+18i}=\frac{x\left(3x+7\right)}{4ix^{2}+18ix+18i}$$

Dividing by 18ix+18i+4ix^{2} undoes the multiplication by 18ix+18i+4ix^{2}.

$$v=\frac{x\left(3x+7\right)}{4ix^{2}+18ix+18i}$$

Divide x\left(7+3x\right) by 18ix+18i+4ix^{2}.

$$v=\frac{x\left(3x+7\right)}{2\left(2ix^{2}+9ix+9i\right)}$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{49-36iv-36v^{2}}+18iv-7}{2\left(3-4iv\right)}\text{; }x=\frac{-\sqrt{49-36iv-36v^{2}}+18iv-7}{2\left(3-4iv\right)}\text{, }&v\neq -\frac{3}{4}i\\x=-\frac{27}{13}\text{, }&v=-\frac{3}{4}i\end{matrix}\right.$$