To solve the inequality \( \log_{0.75}(x + 3) > \log_{0.5}(x + 3) \), we start by recognizing that both logarithmic functions are defined only if \( x + 3 > 0 \), which means \( x > -3 \).
Next, we can rewrite the logarithm into a common base. In this case, we can convert the logarithm to base 10 or natural logarithm for easier comparison. However, notice that since \( 0.75 < 1 \) and \( 0.5 < 1 \), the logarithm will be a decreasing function:
- \( \log_{0.75}(x + 3) = \frac{\log(x + 3)}{\log(0.75)} \)
- \( \log_{0.5}(x + 3) = \frac{\log(x + 3)}{\log(0.5)} \)
Since both logarithmic functions are decreasing, we can find the point where they are equal:
Set: \[ \log_{0.75}(x + 3) = \log_{0.5}(x + 3) \]
Because both sides can be multiplied by the positive value \( \log(0.75) \) and \( \log(0.5) \) (logarithmic bases less than 1 lead to a flip in the inequality):
\[ \log(x + 3) \cdot \log(0.5) < \log(x + 3) \cdot \log(0.75) \]
This means whenever \( \log(x + 3) \) is positive, \( \log_{0.75}(x + 3) \) is greater than \( \log_{0.5}(x + 3) \); hence we need to find \( x + 3 > 1 \) or \( x > -2 \).
So the solution to the inequality is: \[ x > -2 \]
Now remembering the previous condition that \( x + 3 > 0 \) which translates to \( x > -3 \), we find that:
\( x > -2 \) is the more restrictive of the two conditions.
Therefore, the range of values for \( x \) is: \[ \text{negative 2 less than x less than infinity} \]
So the answer is: negative 2 less than x less than infinity.