Solve xy/x^{2-1}divx^2-x-6/x^2+x*x^2-2x-3/x^2y+xy

$$\frac { x y } { x ^ { 2 } - 1 } \div \frac { x ^ { 2 } - x - 6 } { x ^ { 2 } + x } \times \frac { x ^ { 2 } - 2 x - 3 } { x ^ { 2 } y + x y }$$

Evaluate

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

Short Solution Steps

Divide \frac{xy}{x^{2}-1} by \frac{x^{2}-x-6}{x^{2}+x} by multiplying \frac{xy}{x^{2}-1} by the reciprocal of \frac{x^{2}-x-6}{x^{2}+x}.

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

Factor the expressions that are not already factored in \frac{xy\left(x^{2}+x\right)}{\left(x^{2}-1\right)\left(x^{2}-x-6\right)}.

$$\frac{y\left(x+1\right)x^{2}}{\left(x-3\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}\times \frac{x^{2}-2x-3}{x^{2}y+xy}$$

Cancel out x+1 in both numerator and denominator.

$$\frac{yx^{2}}{\left(x-3\right)\left(x-1\right)\left(x+2\right)}\times \frac{x^{2}-2x-3}{x^{2}y+xy}$$

Factor the expressions that are not already factored in \frac{x^{2}-2x-3}{x^{2}y+xy}.

$$\frac{yx^{2}}{\left(x-3\right)\left(x-1\right)\left(x+2\right)}\times \frac{\left(x-3\right)\left(x+1\right)}{xy\left(x+1\right)}$$

Cancel out x+1 in both numerator and denominator.

$$\frac{yx^{2}}{\left(x-3\right)\left(x-1\right)\left(x+2\right)}\times \frac{x-3}{xy}$$

Multiply \frac{yx^{2}}{\left(x-3\right)\left(x-1\right)\left(x+2\right)} times \frac{x-3}{xy} by multiplying numerator times numerator and denominator times denominator.

$$\frac{yx^{2}\left(x-3\right)}{\left(x-3\right)\left(x-1\right)\left(x+2\right)xy}$$

Cancel out xy\left(x-3\right) in both numerator and denominator.

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

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

$$\frac{x}{x^{2}+x-2}$$

Expand

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

Short Solution Steps

Divide \frac{xy}{x^{2}-1} by \frac{x^{2}-x-6}{x^{2}+x} by multiplying \frac{xy}{x^{2}-1} by the reciprocal of \frac{x^{2}-x-6}{x^{2}+x}.

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

Factor the expressions that are not already factored in \frac{xy\left(x^{2}+x\right)}{\left(x^{2}-1\right)\left(x^{2}-x-6\right)}.

$$\frac{y\left(x+1\right)x^{2}}{\left(x-3\right)\left(x-1\right)\left(x+1\right)\left(x+2\right)}\times \frac{x^{2}-2x-3}{x^{2}y+xy}$$

Cancel out x+1 in both numerator and denominator.

$$\frac{yx^{2}}{\left(x-3\right)\left(x-1\right)\left(x+2\right)}\times \frac{x^{2}-2x-3}{x^{2}y+xy}$$

Factor the expressions that are not already factored in \frac{x^{2}-2x-3}{x^{2}y+xy}.

$$\frac{yx^{2}}{\left(x-3\right)\left(x-1\right)\left(x+2\right)}\times \frac{\left(x-3\right)\left(x+1\right)}{xy\left(x+1\right)}$$

Cancel out x+1 in both numerator and denominator.

$$\frac{yx^{2}}{\left(x-3\right)\left(x-1\right)\left(x+2\right)}\times \frac{x-3}{xy}$$

Multiply \frac{yx^{2}}{\left(x-3\right)\left(x-1\right)\left(x+2\right)} times \frac{x-3}{xy} by multiplying numerator times numerator and denominator times denominator.

$$\frac{yx^{2}\left(x-3\right)}{\left(x-3\right)\left(x-1\right)\left(x+2\right)xy}$$

Cancel out xy\left(x-3\right) in both numerator and denominator.

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

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

$$\frac{x}{x^{2}+x-2}$$