To solve the inequality \( \log_{(1/10)}(x^2 + x) > \log_{(1/2)}(x^3 - x) + \log_{2}(x - 1) \), we will follow these steps:
### Step 1: Determine the conditions for the logarithmic expressions to be defined
For the logarithmic expressions to be defined, the arguments of the logarithms must be positive.
1. **For \( \log_{(1/10)}(x^2 + x) \)**:
\[
x^2 + x > 0
\]
Factoring gives:
\[
x(x + 1) > 0
\]
The critical points are \( x = 0 \) and \( x = -1 \). Testing intervals:
- For \( x < -1 \): both factors are negative, product is positive.
- For \( -1 < x < 0 \): one factor is negative, product is negative.
- For \( x > 0 \): both factors are positive, product is positive.
Thus, \( x(x + 1) > 0 \) for \( x \in (-\infty, -1) \cup (0, \infty) \).
2. **For \( \log_{(1/2)}(x^3 - x) \)**:
\[
x^3 - x > 0
\]
Factoring gives:
\[
x(x^2 - 1) > 0 \quad \Rightarrow \quad x(x - 1)(x + 1) > 0
\]
The critical points are \( x = -1, 0, 1 \). Testing intervals:
- For \( x < -1 \): all factors are negative, product is negative.
- For \( -1 < x < 0 \): one factor is negative, product is positive.
- For \( 0 < x < 1 \): one factor is negative, product is negative.
- For \( x > 1 \): all factors are positive, product is positive.
Thus, \( x(x - 1)(x + 1) > 0 \) for \( x \in (-1, 0) \cup (1, \infty) \).
3. **For \( \log_{2}(x - 1) \)**:
\[
x - 1 > 0 \quad \Rightarrow \quad x > 1
\]
### Step 2: Combine the intervals
Now we combine the intervals where all logarithmic expressions are defined:
- From \( \log_{(1/10)}(x^2 + x) \): \( (-\infty, -1) \cup (0, \infty) \)
- From \( \log_{(1/2)}(x^3 - x) \): \( (-1, 0) \cup (1, \infty) \)
- From \( \log_{2}(x - 1) \): \( (1, \infty) \)
The common interval that satisfies all conditions is:
\[
(1, \infty)
\]
### Step 3: Solve the inequality
Now we need to solve the original inequality. Since we have already established that \( x \) must be in \( (1, \infty) \), we can proceed to analyze the inequality within this interval.
### Final Result
The solution to the inequality \( \log_{(1/10)}(x^2 + x) > \log_{(1/2)}(x^3 - x) + \log_{2}(x - 1) \) is:
\[
x \in (1, \infty)
\]
