[tex]\displaystyle\bf\\\frac{3x+2}{2} +\frac{5-2x}{3}=6 \\\\\\\frac{3x+2}{2}\cdot 6 +\frac{5-2x}{3}\cdot 6=6\cdot 6\\\\\\(3x+2)\cdot 3+(5-2x)\cdot 2=36\\\\\\9x+6+10-4x=36\\\\\\5x+16=36\\\\\\5x=36-16\\\\\\5x=20\\\\\\x=20:5\\\\\\\boxed{x=4}[/tex]

Verified answer

[tex] \frac{3x + 2}{2} + \frac{5 - 2x}{3} = 6 \: \: | \times 6 \\ 3(3x + 2) + 2(5 - 2x) = 36 \\ 9x + 6 + 10 - 4x = 36 \\ 5x = 36 - 16 \\ 5x = 20 \\ x = 20 \div 5 \\ x = 4[/tex]