Solve 1/x+1(1/x+3+1/x+2)

$$\frac { 1 } { x + 1 } ( \frac { 1 } { x + 3 } + \frac { 1 } { x + 2 } )$$

Evaluate

$$\frac{2x+5}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+3 and x+2 is \left(x+2\right)\left(x+3\right). Multiply \frac{1}{x+3} times \frac{x+2}{x+2}. Multiply \frac{1}{x+2} times \frac{x+3}{x+3}.

$$\frac{1}{x+1}\left(\frac{x+2}{\left(x+2\right)\left(x+3\right)}+\frac{x+3}{\left(x+2\right)\left(x+3\right)}\right)$$

Since \frac{x+2}{\left(x+2\right)\left(x+3\right)} and \frac{x+3}{\left(x+2\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{1}{x+1}\times \frac{x+2+x+3}{\left(x+2\right)\left(x+3\right)}$$

Combine like terms in x+2+x+3.

$$\frac{1}{x+1}\times \frac{2x+5}{\left(x+2\right)\left(x+3\right)}$$

Multiply \frac{1}{x+1} times \frac{2x+5}{\left(x+2\right)\left(x+3\right)} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2x+5}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}$$

Apply the distributive property by multiplying each term of x+1 by each term of x+2.

$$\frac{2x+5}{\left(x^{2}+2x+x+2\right)\left(x+3\right)}$$

Combine 2x and x to get 3x.

$$\frac{2x+5}{\left(x^{2}+3x+2\right)\left(x+3\right)}$$

Apply the distributive property by multiplying each term of x^{2}+3x+2 by each term of x+3.

$$\frac{2x+5}{x^{3}+3x^{2}+3x^{2}+9x+2x+6}$$

Combine 3x^{2} and 3x^{2} to get 6x^{2}.

$$\frac{2x+5}{x^{3}+6x^{2}+9x+2x+6}$$

Combine 9x and 2x to get 11x.

$$\frac{2x+5}{x^{3}+6x^{2}+11x+6}$$

Expand

$$\frac{2x+5}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+3 and x+2 is \left(x+2\right)\left(x+3\right). Multiply \frac{1}{x+3} times \frac{x+2}{x+2}. Multiply \frac{1}{x+2} times \frac{x+3}{x+3}.

$$\frac{1}{x+1}\left(\frac{x+2}{\left(x+2\right)\left(x+3\right)}+\frac{x+3}{\left(x+2\right)\left(x+3\right)}\right)$$

Since \frac{x+2}{\left(x+2\right)\left(x+3\right)} and \frac{x+3}{\left(x+2\right)\left(x+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{1}{x+1}\times \frac{x+2+x+3}{\left(x+2\right)\left(x+3\right)}$$

Combine like terms in x+2+x+3.

$$\frac{1}{x+1}\times \frac{2x+5}{\left(x+2\right)\left(x+3\right)}$$

Multiply \frac{1}{x+1} times \frac{2x+5}{\left(x+2\right)\left(x+3\right)} by multiplying numerator times numerator and denominator times denominator.

$$\frac{2x+5}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}$$

Apply the distributive property by multiplying each term of x+1 by each term of x+2.

$$\frac{2x+5}{\left(x^{2}+2x+x+2\right)\left(x+3\right)}$$

Combine 2x and x to get 3x.

$$\frac{2x+5}{\left(x^{2}+3x+2\right)\left(x+3\right)}$$

Apply the distributive property by multiplying each term of x^{2}+3x+2 by each term of x+3.

$$\frac{2x+5}{x^{3}+3x^{2}+3x^{2}+9x+2x+6}$$

Combine 3x^{2} and 3x^{2} to get 6x^{2}.

$$\frac{2x+5}{x^{3}+6x^{2}+9x+2x+6}$$

Combine 9x and 2x to get 11x.

$$\frac{2x+5}{x^{3}+6x^{2}+11x+6}$$