The problem asks to write the expression

$\frac{7}{x - 3} - \frac{5}{x + 2}$

as a single fraction in its simplest form.

Steps:

1. Find a common denominator: The denominators are $x - 3$ and $x + 2$. The common denominator will be the product of these two: $(x - 3)(x + 2)$.

2. Rewrite each fraction with the common denominator: $\frac{7}{x - 3} = \frac{7(x + 2)}{(x - 3)(x + 2)}$ $\frac{5}{x + 2} = \frac{5(x - 3)}{(x - 3)(x + 2)}$

3. Perform the subtraction: $\frac{7(x + 2)}{(x - 3)(x + 2)} - \frac{5(x - 3)}{(x - 3)(x + 2)}$

4. Expand the numerators: $7(x + 2) = 7x + 14$ $5(x - 3) = 5x - 15$

5. Subtract the numerators: $(7x + 14) - (5x - 15) = 7x + 14 - 5x + 15 = 2x + 29$

6. Write the final expression: $\frac{2x + 29}{(x - 3)(x + 2)}$

Thus, the simplified single fraction is:

$\frac{2x + 29}{(x - 3)(x + 2)}$

Let me know if you'd like more details on any part of the solution.

Related Questions:

1. What are the steps to factor quadratic expressions?
2. How do you find the least common denominator for rational expressions?
3. What is the difference between multiplying and adding fractions with variables?
4. How can you verify if a rational expression is simplified?
5. What are restrictions on the variable in the final answer?

Tip:

Always check for common factors between the numerator and the denominator after simplifying fractions.