To determine the domain of the function [tex]\( y = \ln \left( \frac{-x + 3}{2} \right) \)[/tex], we must ensure that the argument of the logarithmic function is positive, because the natural logarithm, [tex]\( \ln(z) \)[/tex], is defined only for [tex]\( z > 0 \)[/tex].
Let's start by setting up the inequality:
[tex]\[
\frac{-x + 3}{2} > 0
\][/tex]
To solve this inequality:
1. Multiply both sides of the inequality by 2 to clear the fraction:
[tex]\[
-x + 3 > 0
\][/tex]
2. Next, solve for [tex]\( x \)[/tex] by isolating it on one side of the inequality:
[tex]\[
-x > -3
\][/tex]
3. Divide both sides by -1, and remember to reverse the inequality sign when dividing by a negative number:
[tex]\[
x < 3
\][/tex]
Thus, the domain of the function [tex]\( y = \ln \left( \frac{-x + 3}{2} \right) \)[/tex] is all [tex]\( x \)[/tex] such that [tex]\( x < 3 \)[/tex].
Therefore, the correct answer is:
[tex]\[
x < 3
\][/tex]
