Solve x/x^2-5x+6=6/x^2-x-6+x+1/x^2-4

$$\frac { x } { x ^ { 2 } - 5 x + 6 } = \frac { 6 } { x ^ { 2 } - x - 6 } + \frac { x + 1 } { x ^ { 2 } - 4 }$$

Solve for x

$$x = \frac{15}{2} = 7\frac{1}{2} = 7.5$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -2,2,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-2\right)\left(x+2\right), the least common multiple of x^{2}-5x+6,x^{2}-x-6,x^{2}-4.

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

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

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

Use the distributive property to multiply x-2 by 6.

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

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

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

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

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

Subtract 3 from -12 to get -15.

$$x^{2}+2x=4x-15+x^{2}$$

Subtract 4x from both sides.

$$x^{2}+2x-4x=-15+x^{2}$$

Combine 2x and -4x to get -2x.

$$x^{2}-2x=-15+x^{2}$$

Subtract x^{2} from both sides.

$$x^{2}-2x-x^{2}=-15$$

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

$$-2x=-15$$

Divide both sides by -2.

$$x=\frac{-15}{-2}$$

Fraction \frac{-15}{-2} can be simplified to \frac{15}{2} by removing the negative sign from both the numerator and the denominator.

$$x=\frac{15}{2}$$