Solve 1/x+z-2-1/x-2/z

$$\frac{ \frac{ 1 }{ x+z-2 } - \frac{ 1 }{ x-2 } }{ z }$$

Evaluate

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

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x-2-x-z+2.

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

Express \frac{\frac{-z}{\left(x-2\right)\left(x+z-2\right)}}{z} as a single fraction.

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

Cancel out z in both numerator and denominator.

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

Apply the distributive property by multiplying each term of x-2 by each term of x+z-2.

$$\frac{-1}{x^{2}+xz-2x-2x-2z+4}$$

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

$$\frac{-1}{x^{2}+xz-4x-2z+4}$$

Expand

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

Short Solution Steps

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

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

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

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

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

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

Combine like terms in x-2-x-z+2.

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

Express \frac{\frac{-z}{\left(x-2\right)\left(x+z-2\right)}}{z} as a single fraction.

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

Cancel out z in both numerator and denominator.

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

Apply the distributive property by multiplying each term of x-2 by each term of x+z-2.

$$\frac{-1}{x^{2}+xz-2x-2x-2z+4}$$

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

$$\frac{-1}{x^{2}+xz-4x-2z+4}$$