### Instruction: Simplify $\frac{x+1}{3}+\frac{2-3x}{2}$.  Express your answer as a single fraction.### Response: To simplify this expression, I need to find a common denominator for the two fractions.
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 a denominator of 6.
This gives me $\frac{2x+2}{6}+\frac{6-9x}{6}$.
Now I can combine the fractions by adding the numerators and keeping the denominator the same.
This yields $\frac{2x+2+6-9x}{6}$.
I can simplify the numerator by combining like terms.
I have $2x-9x=-7x$ and $2+6=8$.
So the numerator is $-7x+8$.
The final simplified fraction is $\frac{-7x+8}{6}$.