Solve 5x+2/2(x^2-4)+x/2x+3=x+4/2x+3+1/2(x-2)

$$\frac { 5 x + 2 } { 2 ( x ^ { 2 } - 4 ) } + \frac { x } { 2 x + 3 } = \frac { x + 4 } { 2 x + 3 } + \frac { 1 } { 2 ( x - 2 ) }$$

Solve for x

$$x = -\frac{8}{3} = -2\frac{2}{3} \approx -2.666666667$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -2,-\frac{3}{2},2 since division by zero is not defined. Multiply both sides of the equation by 2\left(x-2\right)\left(x+2\right)\left(2x+3\right), the least common multiple of 2\left(x^{2}-4\right),2x+3,2\left(x-2\right).

$$\left(2x+3\right)\left(5x+2\right)+\left(2x^{2}-8\right)x=\left(2x^{2}-8\right)\left(x+4\right)+\left(x+2\right)\left(2x+3\right)$$

Use the distributive property to multiply 2x+3 by 5x+2 and combine like terms.

$$10x^{2}+19x+6+\left(2x^{2}-8\right)x=\left(2x^{2}-8\right)\left(x+4\right)+\left(x+2\right)\left(2x+3\right)$$

Use the distributive property to multiply 2x^{2}-8 by x.

$$10x^{2}+19x+6+2x^{3}-8x=\left(2x^{2}-8\right)\left(x+4\right)+\left(x+2\right)\left(2x+3\right)$$

Combine 19x and -8x to get 11x.

$$10x^{2}+11x+6+2x^{3}=\left(2x^{2}-8\right)\left(x+4\right)+\left(x+2\right)\left(2x+3\right)$$

Use the distributive property to multiply 2x^{2}-8 by x+4.

$$10x^{2}+11x+6+2x^{3}=2x^{3}+8x^{2}-8x-32+\left(x+2\right)\left(2x+3\right)$$

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

$$10x^{2}+11x+6+2x^{3}=2x^{3}+8x^{2}-8x-32+2x^{2}+7x+6$$

Combine 8x^{2} and 2x^{2} to get 10x^{2}.

$$10x^{2}+11x+6+2x^{3}=2x^{3}+10x^{2}-8x-32+7x+6$$

Combine -8x and 7x to get -x.

$$10x^{2}+11x+6+2x^{3}=2x^{3}+10x^{2}-x-32+6$$

Add -32 and 6 to get -26.

$$10x^{2}+11x+6+2x^{3}=2x^{3}+10x^{2}-x-26$$

Subtract 2x^{3} from both sides.

$$10x^{2}+11x+6+2x^{3}-2x^{3}=10x^{2}-x-26$$

Combine 2x^{3} and -2x^{3} to get 0.

$$10x^{2}+11x+6=10x^{2}-x-26$$

Subtract 10x^{2} from both sides.

$$10x^{2}+11x+6-10x^{2}=-x-26$$

Combine 10x^{2} and -10x^{2} to get 0.

$$11x+6=-x-26$$

Add x to both sides.

$$11x+6+x=-26$$

Combine 11x and x to get 12x.

$$12x+6=-26$$

Subtract 6 from both sides.

$$12x=-26-6$$

Subtract 6 from -26 to get -32.

$$12x=-32$$

Divide both sides by 12.

$$x=\frac{-32}{12}$$

Reduce the fraction \frac{-32}{12} to lowest terms by extracting and canceling out 4.

$$x=-\frac{8}{3}$$