Solve x+1/x-2-x/x+3=x+m/left(x-2right)left(x+3right)

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

Solve for m

$$m=5x+3$$

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

Solution Steps

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-2\right)\left(x+3\right).

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

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

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

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

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

To find the opposite of x^{2}-2x, find the opposite of each term.

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

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

$$4x+3+2x=x+m$$

Combine 4x and 2x to get 6x.

$$6x+3=x+m$$

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

$$x+m=6x+3$$

Subtract x from both sides.

$$m=6x+3-x$$

Combine 6x and -x to get 5x.

$$m=5x+3$$

Solve for x

$$x=\frac{m-3}{5}$$

$$m\neq 13\text{ and }m\neq -12$$

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-2\right)\left(x+3\right).

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

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

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

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

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

To find the opposite of x^{2}-2x, find the opposite of each term.

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

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

$$4x+3+2x=x+m$$

Combine 4x and 2x to get 6x.

$$6x+3=x+m$$

Subtract x from both sides.

$$6x+3-x=m$$

Combine 6x and -x to get 5x.

$$5x+3=m$$

Subtract 3 from both sides.

$$5x=m-3$$

Divide both sides by 5.

$$\frac{5x}{5}=\frac{m-3}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x=\frac{m-3}{5}$$

Variable x cannot be equal to any of the values -3,2.

$$x=\frac{m-3}{5}\text{, }x\neq -3\text{ and }x\neq 2$$