Solve 6/x-2=x/x+3-1

$$\frac{ 6 }{ x-2 } = \frac{ x }{ x+3 } -1$$

Solve for x

$$x = -\frac{4}{3} = -1\frac{1}{3} \approx -1.333333333$$

Steps for Solving Linear Equation

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

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

Use the distributive property to multiply x+3 by 6.

$$6x+18=\left(x-2\right)x+\left(x-2\right)\left(x+3\right)\left(-1\right)$$

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

$$6x+18=x^{2}-2x+\left(x-2\right)\left(x+3\right)\left(-1\right)$$

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

$$6x+18=x^{2}-2x+\left(x^{2}+x-6\right)\left(-1\right)$$

Use the distributive property to multiply x^{2}+x-6 by -1.

$$6x+18=x^{2}-2x-x^{2}-x+6$$

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

$$6x+18=-2x-x+6$$

Combine -2x and -x to get -3x.

$$6x+18=-3x+6$$

Add 3x to both sides.

$$6x+18+3x=6$$

Combine 6x and 3x to get 9x.

$$9x+18=6$$

Subtract 18 from both sides.

$$9x=6-18$$

Subtract 18 from 6 to get -12.

$$9x=-12$$

Divide both sides by 9.

$$x=\frac{-12}{9}$$

Reduce the fraction \frac{-12}{9} to lowest terms by extracting and canceling out 3.

$$x=-\frac{4}{3}$$