We are given the equation:

$4 + \frac{7}{x} = \frac{5}{3}$

Steps to solve for $x$:

1. Subtract 4 from both sides: $\frac{7}{x} = \frac{5}{3} - 4$

   To subtract 4, express it as a fraction with a denominator of 3: $\frac{5}{3} - \frac{12}{3} = \frac{-7}{3}$ Now the equation becomes: $\frac{7}{x} = \frac{-7}{3}$

2. Cross multiply: $7 \cdot 3 = -7 \cdot x$ This simplifies to: $21 = -7x$

3. Solve for $x$: $x = \frac{21}{-7} = -3$

Thus, the solution is $x = -3$.

Would you like any further details on this solution? Here are some related questions:

1. How would you solve this if the equation had decimals instead of fractions?
2. How do we check if the solution is correct?
3. Can this approach be applied to other similar rational equations?
4. What happens if the variable appears in both denominators?
5. What types of equations require cross multiplication for solution?

Tip: Always simplify fractions and expressions before solving equations—it often makes the calculations easier!