Answer :
To find the domain of the function [tex]\( y = \ln \left( \frac{-x+3}{2} \right) \)[/tex], we must ensure that the argument of the natural logarithm is positive. Specifically, the expression [tex]\( \ln(a) \)[/tex] is only defined for [tex]\( a > 0 \)[/tex].
Given our function [tex]\( y = \ln \left( \frac{-x+3}{2} \right) \)[/tex], we need:
[tex]\[ \frac{-x+3}{2} > 0 \][/tex]
To solve this inequality:
1. Start by isolating the fraction:
[tex]\[ \frac{-x+3}{2} > 0 \][/tex]
2. Multiply both sides of the inequality by 2 to clear the denominator:
[tex]\[ -x + 3 > 0 \][/tex]
3. Now, solve for [tex]\( x \)[/tex] by isolating [tex]\( x \)[/tex]:
[tex]\[ -x + 3 > 0 \][/tex]
[tex]\[ -x > -3 \][/tex]
4. Divide both sides by [tex]\(-1\)[/tex]. Remember, dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign:
[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] values such that [tex]\( x < 3 \)[/tex].
Therefore, the correct answer is:
[tex]\[ x < 3 \][/tex]
Given our function [tex]\( y = \ln \left( \frac{-x+3}{2} \right) \)[/tex], we need:
[tex]\[ \frac{-x+3}{2} > 0 \][/tex]
To solve this inequality:
1. Start by isolating the fraction:
[tex]\[ \frac{-x+3}{2} > 0 \][/tex]
2. Multiply both sides of the inequality by 2 to clear the denominator:
[tex]\[ -x + 3 > 0 \][/tex]
3. Now, solve for [tex]\( x \)[/tex] by isolating [tex]\( x \)[/tex]:
[tex]\[ -x + 3 > 0 \][/tex]
[tex]\[ -x > -3 \][/tex]
4. Divide both sides by [tex]\(-1\)[/tex]. Remember, dividing or multiplying both sides of an inequality by a negative number reverses the inequality sign:
[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] values such that [tex]\( x < 3 \)[/tex].
Therefore, the correct answer is:
[tex]\[ x < 3 \][/tex]
