### Instruction: Simplify $\frac{x+1}{3}+\frac{2-3x}{2}$.  Express your answer as a single fraction.### Response: To simplify this expression, I want to combine the two fractions into one by finding a common denominator.
The least common multiple of 3 and 2 is 6, so I can multiply the first fraction by $\frac{2}{2}$ and the second fraction by $\frac{3}{3}$ to get equivalent fractions with denominator 6.
This gives me $\frac{2x+2}{6}+\frac{6-9x}{6}$.
Now I can add the numerators and keep the denominator, since the fractions have the same denominator.
This gives me $\frac{2x+2+6-9x}{6}$, which I can simplify by combining like terms.
This gives me $\frac{-7x+8}{6}$, which is the simplest form of the fraction.