### Instruction: Simplify $\frac{x+1}{3}+\frac{2-3x}{2}$.  Express your answer as a single fraction.### Response: I need to add two fractions with different denominators, so I have to find a common denominator first.
The lowest common denominator for 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 the same denominator.
That gives me $\frac{2(x+1)}{6}+\frac{3(2-3x)}{6}$.
Now I can add the numerators and keep the denominator, since they are the same.
That gives me $\frac{2(x+1)+3(2-3x)}{6}$.
I can simplify the numerator by distributing the 2 and the 3 and combining like terms.
That gives me $\frac{2x+2+6-9x}{6}$, which simplifies to $\frac{-7x+8}{6}$.
This is the simplest form of the fraction, so I'm done.