Solve 2/n+2+2/2x-1=3/n+1

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

Solve for x

$$x=\frac{2n^{2}+7n+8}{2\left(n+4\right)}$$

$$n\neq -2\text{ and }n\neq -1\text{ and }n\neq -4$$

Steps for Solving Linear Equation

Variable x cannot be equal to \frac{1}{2} since division by zero is not defined. Multiply both sides of the equation by \left(2x-1\right)\left(n+1\right)\left(n+2\right), the least common multiple of n+2,2x-1,n+1.

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

Use the distributive property to multiply 2x-1 by n+1.

$$\left(2xn+2x-n-1\right)\times 2+\left(n+1\right)\left(n+2\right)\times 2=\left(2x-1\right)\left(n+2\right)\times 3$$

Use the distributive property to multiply 2xn+2x-n-1 by 2.

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

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

$$4nx+4x-2n-2+\left(n^{2}+3n+2\right)\times 2=\left(2x-1\right)\left(n+2\right)\times 3$$

Use the distributive property to multiply n^{2}+3n+2 by 2.

$$4nx+4x-2n-2+2n^{2}+6n+4=\left(2x-1\right)\left(n+2\right)\times 3$$

Combine -2n and 6n to get 4n.

$$4nx+4x+4n-2+2n^{2}+4=\left(2x-1\right)\left(n+2\right)\times 3$$

Add -2 and 4 to get 2.

$$4nx+4x+4n+2+2n^{2}=\left(2x-1\right)\left(n+2\right)\times 3$$

Use the distributive property to multiply 2x-1 by n+2.

$$4nx+4x+4n+2+2n^{2}=\left(2xn+4x-n-2\right)\times 3$$

Use the distributive property to multiply 2xn+4x-n-2 by 3.

$$4nx+4x+4n+2+2n^{2}=6nx+12x-3n-6$$

Subtract 6nx from both sides.

$$4nx+4x+4n+2+2n^{2}-6nx=12x-3n-6$$

Combine 4nx and -6nx to get -2nx.

$$-2nx+4x+4n+2+2n^{2}=12x-3n-6$$

Subtract 12x from both sides.

$$-2nx+4x+4n+2+2n^{2}-12x=-3n-6$$

Combine 4x and -12x to get -8x.

$$-2nx-8x+4n+2+2n^{2}=-3n-6$$

Subtract 4n from both sides.

$$-2nx-8x+2+2n^{2}=-3n-6-4n$$

Combine -3n and -4n to get -7n.

$$-2nx-8x+2+2n^{2}=-7n-6$$

Subtract 2 from both sides.

$$-2nx-8x+2n^{2}=-7n-6-2$$

Subtract 2 from -6 to get -8.

$$-2nx-8x+2n^{2}=-7n-8$$

Subtract 2n^{2} from both sides.

$$-2nx-8x=-7n-8-2n^{2}$$

Combine all terms containing x.

$$\left(-2n-8\right)x=-7n-8-2n^{2}$$

The equation is in standard form.

$$\left(-2n-8\right)x=-2n^{2}-7n-8$$

Divide both sides by -2n-8.

$$\frac{\left(-2n-8\right)x}{-2n-8}=\frac{-2n^{2}-7n-8}{-2n-8}$$

Dividing by -2n-8 undoes the multiplication by -2n-8.

$$x=\frac{-2n^{2}-7n-8}{-2n-8}$$

Divide -8-7n-2n^{2} by -2n-8.

$$x=\frac{2n^{2}+7n+8}{2\left(n+4\right)}$$

Variable x cannot be equal to \frac{1}{2}.

$$x=\frac{2n^{2}+7n+8}{2\left(n+4\right)}\text{, }x\neq \frac{1}{2}$$

Solve for n (complex solution)

$$n=\frac{\sqrt{4x^{2}+36x-15}}{4}+\frac{x}{2}-\frac{7}{4}$$

$$n=-\frac{\sqrt{4x^{2}+36x-15}}{4}+\frac{x}{2}-\frac{7}{4}\text{, }x\neq \frac{1}{2}$$

Solve for n

$$n=\frac{\sqrt{4x^{2}+36x-15}}{4}+\frac{x}{2}-\frac{7}{4}$$

$$n=-\frac{\sqrt{4x^{2}+36x-15}}{4}+\frac{x}{2}-\frac{7}{4}\text{, }x\leq -2\sqrt{6}-\frac{9}{2}\text{ or }\left(x\neq \frac{1}{2}\text{ and }x\geq 2\sqrt{6}-\frac{9}{2}\right)$$