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

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

Evaluate

$$-\frac{3}{\left(x+2\right)\left(x+h+2\right)}$$

Short Solution Steps

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

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

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

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

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

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

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

$$\frac{\frac{-3h}{\left(x+2\right)\left(x+h+2\right)}}{h}$$

Express \frac{\frac{-3h}{\left(x+2\right)\left(x+h+2\right)}}{h} as a single fraction.

$$\frac{-3h}{\left(x+2\right)\left(x+h+2\right)h}$$

Cancel out h in both numerator and denominator.

$$\frac{-3}{\left(x+2\right)\left(x+h+2\right)}$$

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

$$\frac{-3}{x^{2}+xh+2x+2x+2h+4}$$

Combine 2x and 2x to get 4x.

$$\frac{-3}{x^{2}+xh+4x+2h+4}$$

Expand

$$-\frac{3}{\left(x+2\right)\left(x+h+2\right)}$$

Short Solution Steps

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

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

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

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

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

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

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

$$\frac{\frac{-3h}{\left(x+2\right)\left(x+h+2\right)}}{h}$$

Express \frac{\frac{-3h}{\left(x+2\right)\left(x+h+2\right)}}{h} as a single fraction.

$$\frac{-3h}{\left(x+2\right)\left(x+h+2\right)h}$$

Cancel out h in both numerator and denominator.

$$\frac{-3}{\left(x+2\right)\left(x+h+2\right)}$$

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

$$\frac{-3}{x^{2}+xh+2x+2x+2h+4}$$

Combine 2x and 2x to get 4x.

$$\frac{-3}{x^{2}+xh+4x+2h+4}$$