Solve 5x+6/x^2-4-x/x^2-4:x/x-2-x+2/x-2

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

Evaluate

$$-1$$

Short Solution Steps

Divide \frac{x}{x^{2}-4} by \frac{x}{x-2} by multiplying \frac{x}{x^{2}-4} by the reciprocal of \frac{x}{x-2}.

$$\frac{5x+6}{x^{2}-4}-\frac{x\left(x-2\right)}{\left(x^{2}-4\right)x}-\frac{x+2}{x-2}$$

Cancel out x in both numerator and denominator.

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

Factor the expressions that are not already factored in \frac{x-2}{x^{2}-4}.

$$\frac{5x+6}{x^{2}-4}-\frac{x-2}{\left(x-2\right)\left(x+2\right)}-\frac{x+2}{x-2}$$

Cancel out x-2 in both numerator and denominator.

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

Factor x^{2}-4.

$$\frac{5x+6}{\left(x-2\right)\left(x+2\right)}-\frac{1}{x+2}-\frac{x+2}{x-2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x+2\right) and x+2 is \left(x-2\right)\left(x+2\right). Multiply \frac{1}{x+2} times \frac{x-2}{x-2}.

$$\frac{5x+6}{\left(x-2\right)\left(x+2\right)}-\frac{x-2}{\left(x-2\right)\left(x+2\right)}-\frac{x+2}{x-2}$$

Since \frac{5x+6}{\left(x-2\right)\left(x+2\right)} and \frac{x-2}{\left(x-2\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{5x+6-\left(x-2\right)}{\left(x-2\right)\left(x+2\right)}-\frac{x+2}{x-2}$$

Do the multiplications in 5x+6-\left(x-2\right).

$$\frac{5x+6-x+2}{\left(x-2\right)\left(x+2\right)}-\frac{x+2}{x-2}$$

Combine like terms in 5x+6-x+2.

$$\frac{4x+8}{\left(x-2\right)\left(x+2\right)}-\frac{x+2}{x-2}$$

Factor the expressions that are not already factored in \frac{4x+8}{\left(x-2\right)\left(x+2\right)}.

$$\frac{4\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}-\frac{x+2}{x-2}$$

Cancel out x+2 in both numerator and denominator.

$$\frac{4}{x-2}-\frac{x+2}{x-2}$$

Since \frac{4}{x-2} and \frac{x+2}{x-2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4-\left(x+2\right)}{x-2}$$

Do the multiplications in 4-\left(x+2\right).

$$\frac{4-x-2}{x-2}$$

Combine like terms in 4-x-2.

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

Extract the negative sign in 2-x.

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

Cancel out x-2 in both numerator and denominator.

$$-1$$

Factor

$$-1$$