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

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

Step 1

Simplify the denominator.

To write $\frac{1}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

$\frac{1}{\frac{1}{x + 2} \cdot \frac{x + 3}{x + 3} + \frac{1}{x + 3}}$

To write $\frac{1}{x + 3}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{1}{\frac{1}{x + 2} \cdot \frac{x + 3}{x + 3} + \frac{1}{x + 3} \cdot \frac{x + 2}{x + 2}}$

Write each expression with a common denominator of $(x + 2) (x + 3)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{x + 2}$ by $\frac{x + 3}{x + 3}$ .

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

Multiply $\frac{1}{x + 3}$ by $\frac{x + 2}{x + 2}$ .

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

Reorder the factors of $(x + 3) (x + 2)$ .

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

Combine the numerators over the common denominator.

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

Reorder terms.

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

Rewrite $\frac{x + x + 3 + 2}{(x + 2) (x + 3)}$ in a factored form.

Add $x$ and $x$ .

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

Add $3$ and $2$ .

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

Multiply $\frac{(x + 2) (x + 3)}{2 x + 5}$ by $1$ .

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