Simplify the following expression: $\frac{\frac{2}{m} - 1}{\frac{m+7}{m-1} - \frac{3}{m}}$

1. Problem Description

Simplify the following expression:

$\frac{\frac{2}{m} - 1}{\frac{m+7}{m-1} - \frac{3}{m}}$

2. Solution Steps

First, we simplify the numerator:

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

Next, we simplify the denominator:

$\frac{m+7}{m-1} - \frac{3}{m} = \frac{(m+7)m}{(m-1)m} - \frac{3(m-1)}{m(m-1)} = \frac{m^2+7m - 3m + 3}{m(m-1)} = \frac{m^2 + 4m + 3}{m(m-1)}$

We can factor the quadratic in the numerator of the denominator:

$m^2 + 4m + 3 = (m+1)(m+3)$

So the denominator becomes

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

Now we have

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

Thus, the expression simplifies to:

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

3. Final Answer

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