### Instruction: For $y=\frac{1-x}{2x+3}$ and $x\neq-\frac{3}{2}$, what is the value of $y$ that is not attainable? Express your answer as a common fraction.### Response: I notice that the problem asks for a value of $y$ that is not attainable, which means that there is no $x$ such that $y=\frac{1-x}{2x+3}$.
This suggests that there might be a horizontal asymptote for the graph of this function, which is a line that the function approaches but never reaches as $x$ goes to positive or negative infinity.
To find the horizontal asymptote, I can compare the degrees of the numerator and denominator of the rational function.
The degree of the numerator is 1, since the highest power of $x$ is 1.
The degree of the denominator is also 1, since the highest power of $x$ is also 1.
Since the degrees are equal, the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator.
The leading coefficient of the numerator is -1, and the leading coefficient of the denominator is 2.
So the horizontal asymptote is $y=\frac{-1}{2}$.
This means that as $x$ goes to positive or negative infinity, $y$ gets closer and closer to $\frac{-1}{2}$, but never reaches it.
Therefore, the value of $y$ that is not attainable is $\frac{-1}{2}$.