The problem is to simplify the expression $\frac{2}{3}(3x+2) - \frac{3}{4}(12x-3)$.

1. Problem Description

The problem is to simplify the expression $\frac{2}{3}(3x+2) - \frac{3}{4}(12x-3)$.

2. Solution Steps

First, we distribute the fractions to the terms within the parentheses.

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

$-\frac{3}{4}(12x-3) = -\frac{3}{4}(12x) - \frac{3}{4}(-3) = -9x + \frac{9}{4}$

Now, we combine the two expressions:

$2x + \frac{4}{3} - 9x + \frac{9}{4}$

Combine the $x$ terms and the constant terms:

$(2x - 9x) + (\frac{4}{3} + \frac{9}{4})$

$-7x + (\frac{4}{3} + \frac{9}{4})$

To add the fractions, we need a common denominator, which is 12:

$\frac{4}{3} = \frac{4 \times 4}{3 \times 4} = \frac{16}{12}$

$\frac{9}{4} = \frac{9 \times 3}{4 \times 3} = \frac{27}{12}$

So, $\frac{16}{12} + \frac{27}{12} = \frac{16+27}{12} = \frac{43}{12}$

Therefore, the simplified expression is:

$-7x + \frac{43}{12}$

3. Final Answer

$-7x + \frac{43}{12}$