The mathematical expression in the image can be interpreted as:

$\frac{2a - 3b}{a} + \frac{3a + 4b}{2a}$

Let's break this down and simplify it step-by-step.

Step 1: Simplify each term separately.

• For the first term $\frac{2a - 3b}{a}$: $\frac{2a - 3b}{a} = \frac{2a}{a} - \frac{3b}{a} = 2 - \frac{3b}{a}$
• For the second term $\frac{3a + 4b}{2a}$: $\frac{3a + 4b}{2a} = \frac{3a}{2a} + \frac{4b}{2a} = \frac{3}{2} + \frac{2b}{a}$

Step 2: Combine the results.

Now, combine the simplified terms: $2 - \frac{3b}{a} + \frac{3}{2} + \frac{2b}{a}$

Step 3: Combine like terms.

We can now combine the constants and the terms involving $b$: $\left(2 + \frac{3}{2}\right) + \left(-\frac{3b}{a} + \frac{2b}{a}\right)$

• Combine the constants $2 + \frac{3}{2} = \frac{4}{2} + \frac{3}{2} = \frac{7}{2}$.
• Combine the terms involving $b$: $-\frac{3b}{a} + \frac{2b}{a} = -\frac{b}{a}$.

Final simplified result:

$\frac{7}{2} - \frac{b}{a}$

Thus, the expression simplifies to: $\boxed{\frac{7}{2} - \frac{b}{a}}$

Would you like more details on any of the steps or have any questions?

5 Questions for Further Exploration:

1. How would the expression change if the second fraction were $\frac{4a + 5b}{2a}$ instead?
2. Can you simplify $\frac{4a - 6b}{2a} + \frac{5a + 7b}{3a}$?
3. What happens to the expression if $a = 1$ and $b = 2$?
4. How would you handle the case where $a = 0$ in any of these terms?
5. How can you combine fractions with different denominators in algebraic expressions?

Tip:

When simplifying expressions involving fractions, always look for common factors in the numerator and denominator to make the calculations easier!