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

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

Solve for g

$$g=\frac{7x+5}{2\left(6x^{2}+7x+2\right)}$$

$$x\neq -\frac{1}{2}\text{ and }x\neq -\frac{2}{3}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(2x+1\right)\left(3x+2\right), the least common multiple of 2x+1,3x+2.

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

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

$$9x+6-\left(2x+1\right)=2g\left(2x+1\right)\left(3x+2\right)$$

To find the opposite of 2x+1, find the opposite of each term.

$$9x+6-2x-1=2g\left(2x+1\right)\left(3x+2\right)$$

Combine 9x and -2x to get 7x.

$$7x+6-1=2g\left(2x+1\right)\left(3x+2\right)$$

Subtract 1 from 6 to get 5.

$$7x+5=2g\left(2x+1\right)\left(3x+2\right)$$

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

$$7x+5=\left(4xg+2g\right)\left(3x+2\right)$$

Use the distributive property to multiply 4xg+2g by 3x+2 and combine like terms.

$$7x+5=12gx^{2}+14gx+4g$$

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

$$12gx^{2}+14gx+4g=7x+5$$

Combine all terms containing g.

$$\left(12x^{2}+14x+4\right)g=7x+5$$

Divide both sides by 12x^{2}+14x+4.

$$\frac{\left(12x^{2}+14x+4\right)g}{12x^{2}+14x+4}=\frac{7x+5}{12x^{2}+14x+4}$$

Dividing by 12x^{2}+14x+4 undoes the multiplication by 12x^{2}+14x+4.

$$g=\frac{7x+5}{12x^{2}+14x+4}$$

Divide 7x+5 by 12x^{2}+14x+4.

$$g=\frac{7x+5}{2\left(6x^{2}+7x+2\right)}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{\sqrt{4g^{2}+44g+49}+14g-7}{24g}\text{; }x=-\frac{-\sqrt{4g^{2}+44g+49}+14g-7}{24g}\text{, }&g\neq 0\\x=-\frac{5}{7}\text{, }&g=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=-\frac{\sqrt{4g^{2}+44g+49}+14g-7}{24g}\text{; }x=-\frac{-\sqrt{4g^{2}+44g+49}+14g-7}{24g}\text{, }&g\leq -3\sqrt{2}-\frac{11}{2}\text{ or }\left(g\neq 0\text{ and }g\geq 3\sqrt{2}-\frac{11}{2}\right)\\x=-\frac{5}{7}\text{, }&g=0\end{matrix}\right.$$