The equation in the image involves simplifying a complex algebraic expression. Let's break it down step by step to solve and simplify it.

The given equation is:

$\frac{\frac{1}{x+7} - \frac{1}{x}}{7} \cdot \frac{x(x+7)}{x(x+7)} = \frac{x(x+7) \cdot \left( \frac{1}{x+7} - \frac{1}{x} \right)}{7x(x+7)}$

We need to simplify the expression step by step:

Step 1: Simplify the difference inside the parentheses.

We are given $\frac{1}{x+7} - \frac{1}{x}$.

To subtract these, we need a common denominator:

$\frac{1}{x+7} - \frac{1}{x} = \frac{x - (x+7)}{x(x+7)} = \frac{x - x - 7}{x(x+7)} = \frac{-7}{x(x+7)}$

Step 2: Substitute this result back into the equation.

The equation now becomes:

$\frac{\frac{-7}{x(x+7)}}{7} \cdot \frac{x(x+7)}{x(x+7)}$

Step 3: Simplify the expression.

The first part is:

$\frac{\frac{-7}{x(x+7)}}{7} = \frac{-7}{7x(x+7)} = \frac{-1}{x(x+7)}$

Now multiplying by $\frac{x(x+7)}{x(x+7)}$:

$\frac{-1}{x(x+7)} \cdot \frac{x(x+7)}{x(x+7)} = \frac{-1}{x(x+7)}$

Thus, the simplified result is:

$\boxed{\frac{-1}{x(x+7)}}$

This is the final simplified expression.

Would you like further clarification or any more details on this problem?

Here are 5 related questions for further practice:

1. How would you subtract $\frac{1}{x+5} - \frac{1}{x}$?
2. Can you solve $\frac{1}{a+3} + \frac{1}{a}$?
3. Simplify $\frac{2}{y+4} - \frac{3}{y}$.
4. How do you simplify $\frac{x^2}{x+2} - \frac{1}{x+2}$?
5. What happens when you multiply $\frac{1}{x+3} \cdot \frac{1}{x-3}$?

Tip: Always find a common denominator when adding or subtracting fractions with different denominators.