Solve 2(3c-2)/x^2-4+3/2x-1=1

$$\frac { 2 ( 3 c - 2 ) } { x ^ { 2 } - 4 } + \frac { 3 } { 2 x - 1 } = 1$$

Solve for c

$$c=-\frac{-x^{3}+2x^{2}-6}{3\left(2x-1\right)}$$

$$x\neq \frac{1}{2}\text{ and }|x|\neq 2$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(x-2\right)\left(2x-1\right)\left(x+2\right), the least common multiple of x^{2}-4,2x-1.

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

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

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

Use the distributive property to multiply 4x-2 by 3c-2.

$$12xc-8x-6c+4+\left(x^{2}-4\right)\times 3=\left(x-2\right)\left(2x-1\right)\left(x+2\right)$$

Use the distributive property to multiply x^{2}-4 by 3.

$$12xc-8x-6c+4+3x^{2}-12=\left(x-2\right)\left(2x-1\right)\left(x+2\right)$$

Subtract 12 from 4 to get -8.

$$12xc-8x-6c-8+3x^{2}=\left(x-2\right)\left(2x-1\right)\left(x+2\right)$$

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

$$12xc-8x-6c-8+3x^{2}=\left(2x^{2}-5x+2\right)\left(x+2\right)$$

Use the distributive property to multiply 2x^{2}-5x+2 by x+2 and combine like terms.

$$12xc-8x-6c-8+3x^{2}=2x^{3}-x^{2}-8x+4$$

Add 8x to both sides.

$$12xc-6c-8+3x^{2}=2x^{3}-x^{2}-8x+4+8x$$

Combine -8x and 8x to get 0.

$$12xc-6c-8+3x^{2}=2x^{3}-x^{2}+4$$

Add 8 to both sides.

$$12xc-6c+3x^{2}=2x^{3}-x^{2}+4+8$$

Add 4 and 8 to get 12.

$$12xc-6c+3x^{2}=2x^{3}-x^{2}+12$$

Subtract 3x^{2} from both sides.

$$12xc-6c=2x^{3}-x^{2}+12-3x^{2}$$

Combine -x^{2} and -3x^{2} to get -4x^{2}.

$$12xc-6c=2x^{3}-4x^{2}+12$$

Combine all terms containing c.

$$\left(12x-6\right)c=2x^{3}-4x^{2}+12$$

Divide both sides by 12x-6.

$$\frac{\left(12x-6\right)c}{12x-6}=\frac{2x^{3}-4x^{2}+12}{12x-6}$$

Dividing by 12x-6 undoes the multiplication by 12x-6.

$$c=\frac{2x^{3}-4x^{2}+12}{12x-6}$$

Divide 2x^{3}-4x^{2}+12 by 12x-6.

$$c=\frac{x^{3}-2x^{2}+6}{3\left(2x-1\right)}$$