To solve the inequality [tex]\(\log_{1/2}(x-3) > \log_{1/2} 4\)[/tex], we need to understand how logarithms with a base between 0 and 1 behave.
Step-by-Step Solution:
1. Change of Base and Understanding the Inequality:
Since the base of our logarithm [tex]\(\log_{1/2}\)[/tex] is less than 1, the inequality reverses. That is, if [tex]\( \log_{1/2} A > \log_{1/2} B \)[/tex], it means [tex]\( A < B \)[/tex].
Given:
[tex]\[
\log_{1/2}(x-3) > \log_{1/2} 4
\][/tex]
This can be rewritten as:
[tex]\[
x - 3 < 4
\][/tex]
2. Solving the Inequality:
[tex]\[
x - 3 < 4
\][/tex]
Adding 3 to both sides:
[tex]\[
x < 7
\][/tex]
3. Determining the Range:
To ensure that the expression inside the logarithm is defined and real, we need [tex]\( x - 3 \)[/tex] to be greater than 0 because the logarithm of a non-positive number is undefined.
[tex]\[
x - 3 > 0
\][/tex]
Adding 3 to both sides:
[tex]\[
x > 3
\][/tex]
4. Combining the Results:
From these two inequalities, we combine the conditions to get:
[tex]\[
3 < x < 7
\][/tex]
Hence, [tex]\( x \in (3, 7) \)[/tex].
5. Finding [tex]\( b - a \)[/tex]:
Here, [tex]\( a = 3 \)[/tex] and [tex]\( b = 7 \)[/tex].
Thus,
[tex]\[
b - a = 7 - 3 = 4
\][/tex]
Therefore, [tex]\(\boxed{4}\)[/tex] is the value of [tex]\( b - a \)[/tex].
