Solve x+h-2/x+h-1-x-2/x-1/h

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

Evaluate

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

Short Solution Steps

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

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

Since \frac{\left(x+h-2\right)\left(x-1\right)}{\left(x-1\right)\left(x+h-1\right)} and \frac{\left(x-2\right)\left(x+h-1\right)}{\left(x-1\right)\left(x+h-1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x+h-2\right)\left(x-1\right)-\left(x-2\right)\left(x+h-1\right).

$$\frac{\frac{x^{2}-x+hx-h-2x+2-x^{2}-xh+x+2x+2h-2}{\left(x-1\right)\left(x+h-1\right)}}{h}$$

Combine like terms in x^{2}-x+hx-h-2x+2-x^{2}-xh+x+2x+2h-2.

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

Express \frac{\frac{h}{\left(x-1\right)\left(x+h-1\right)}}{h} as a single fraction.

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

Cancel out h in both numerator and denominator.

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

Apply the distributive property by multiplying each term of x-1 by each term of x+h-1.

$$\frac{1}{x^{2}+xh-x-x-h+1}$$

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

$$\frac{1}{x^{2}+xh-2x-h+1}$$

Expand

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

Short Solution Steps

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

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

Since \frac{\left(x+h-2\right)\left(x-1\right)}{\left(x-1\right)\left(x+h-1\right)} and \frac{\left(x-2\right)\left(x+h-1\right)}{\left(x-1\right)\left(x+h-1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(x+h-2\right)\left(x-1\right)-\left(x-2\right)\left(x+h-1\right).

$$\frac{\frac{x^{2}-x+hx-h-2x+2-x^{2}-xh+x+2x+2h-2}{\left(x-1\right)\left(x+h-1\right)}}{h}$$

Combine like terms in x^{2}-x+hx-h-2x+2-x^{2}-xh+x+2x+2h-2.

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

Express \frac{\frac{h}{\left(x-1\right)\left(x+h-1\right)}}{h} as a single fraction.

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

Cancel out h in both numerator and denominator.

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

Apply the distributive property by multiplying each term of x-1 by each term of x+h-1.

$$\frac{1}{x^{2}+xh-x-x-h+1}$$

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

$$\frac{1}{x^{2}+xh-2x-h+1}$$