To simplify the given expression using cross multiplication, we first rewrite it and work step by step:

The expression is:

$\frac{3(x+1)}{2} + \frac{4(x-2)}{3}$

Step 1: Find the least common denominator (LCD)

The denominators are $2$ and $3$. Their least common denominator (LCD) is $6$.

Step 2: Rewrite the fractions with the LCD

Multiply the numerator and denominator of each fraction to make their denominators $6$:

$\frac{3(x+1)}{2} = \frac{3(x+1) \cdot 3}{2 \cdot 3} = \frac{9(x+1)}{6}$ $\frac{4(x-2)}{3} = \frac{4(x-2) \cdot 2}{3 \cdot 2} = \frac{8(x-2)}{6}$

Now the expression becomes:

$\frac{9(x+1)}{6} + \frac{8(x-2)}{6}$

Step 3: Combine the fractions

Since the denominators are the same, we can combine the numerators:

$\frac{9(x+1) + 8(x-2)}{6}$

Step 4: Expand the numerator

Expand $9(x+1)$ and $8(x-2)$:

$9(x+1) = 9x + 9$ $8(x-2) = 8x - 16$

Substitute these into the numerator:

$\frac{(9x + 9) + (8x - 16)}{6}$

Simplify the numerator:

$9x + 8x + 9 - 16 = 17x - 7$

So, the expression becomes:

$\frac{17x - 7}{6}$

Final Answer:

$\frac{17x - 7}{6}$

---

Do you have any questions or need more details?

Here are 5 related questions to expand your understanding:

1. How can you simplify similar expressions with different denominators?
2. What is the importance of finding the least common denominator (LCD)?
3. How would this change if subtraction were involved instead of addition?
4. How can you verify your simplification is correct?
5. Can this expression be solved further for specific values of $x$?

Tip: Always ensure to simplify fractions fully after combining terms!