To solve the inequality \(\frac{4x - 6}{3} \leq \frac{3 + 2x}{2}\), we first get rid of the fractions by multiplying both sides of the inequality by 6 (the least common multiple of 2 and 3) to clear the denominators.

This gives us \(2(4x - 6) \leq 3(3 + 2x)\).

\(8x - 12 \leq 9 + 6x\).

Now, isolate x on one side of the inequality by moving 6x to the left side and 9 to the right side:

\(8x - 6x \leq 9 + 12\).

\(2x \leq 21\).

Finally, divide both sides by 2 to find \(x \leq \frac{21}{2}\) or \(x \leq 10\frac{1}{2}\).