Simplify ((m-2)/(m+1)+2)/(1+(1+2m)/(m+1))

$\frac{\frac{m - 2}{m + 1} + 2}{1 + \frac{1 + 2 m}{m + 1}}$

Step 1

Multiply the numerator and denominator of the complex fraction by $m + 1$ .

Multiply $\frac{\frac{m - 2}{m + 1} + 2}{1 + \frac{1 + 2 m}{m + 1}}$ by $\frac{m + 1}{m + 1}$ .

$\frac{m + 1}{m + 1} \cdot \frac{\frac{m - 2}{m + 1} + 2}{1 + \frac{1 + 2 m}{m + 1}}$

Combine.

$\frac{(m + 1) (\frac{m - 2}{m + 1} + 2)}{(m + 1) (1 + \frac{1 + 2 m}{m + 1})}$

Step 2

Apply the distributive property.

$\frac{(m + 1) \frac{m - 2}{m + 1} + (m + 1) \cdot 2}{(m + 1) \cdot 1 + (m + 1) \frac{1 + 2 m}{m + 1}}$

Step 3

Simplify by cancelling.

Cancel the common factor of $m + 1$ .

Cancel the common factor.

$\frac{(m + 1) \frac{m - 2}{m + 1} + (m + 1) \cdot 2}{(m + 1) \cdot 1 + (m + 1) \frac{1 + 2 m}{m + 1}}$

Rewrite the expression.

$\frac{m - 2 + (m + 1) \cdot 2}{(m + 1) \cdot 1 + (m + 1) \frac{1 + 2 m}{m + 1}}$

Cancel the common factor of $m + 1$ .

Cancel the common factor.

$\frac{m - 2 + (m + 1) \cdot 2}{(m + 1) \cdot 1 + (m + 1) \frac{1 + 2 m}{m + 1}}$

Rewrite the expression.

$\frac{m - 2 + (m + 1) \cdot 2}{(m + 1) \cdot 1 + 1 + 2 m}$

Step 4

Simplify the numerator.

Apply the distributive property.

$\frac{m - 2 + m \cdot 2 + 1 \cdot 2}{(m + 1) \cdot 1 + 1 + 2 m}$

Move $2$ to the left of $m$ .

$\frac{m - 2 + 2 \cdot m + 1 \cdot 2}{(m + 1) \cdot 1 + 1 + 2 m}$

Multiply $2$ by $1$ .

$\frac{m - 2 + 2 \cdot m + 2}{(m + 1) \cdot 1 + 1 + 2 m}$

Add $m$ and $2 m$ .

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

Add $- 2$ and $2$ .

$\frac{3 m + 0}{(m + 1) \cdot 1 + 1 + 2 m}$

Add $3 m$ and $0$ .

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

Step 5

Simplify the denominator.

Multiply $m + 1$ by $1$ .

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

Add $m$ and $2 m$ .

$\frac{3 m}{3 m + 1 + 1}$

Add $1$ and $1$ .

$\frac{3 m}{3 m + 2}$