The equation `(log)_(x+1)(x- .5)=(log)_(x-0. 5)(x+1)` has (A) two real solutions (B) no prime solution (C) one integral solution (D) no irrational solution

Explanation

To solve the equation \( \log_{(x+1)}(x-0.5) = \log_{(x-0.5)}(x+1) \), we will follow these steps:

### Step 1: Rewrite the logarithmic equation
We start with the equation:
\[ \log_{(x+1)}(x-0.5) = \log_{(x-0.5)}(x+1) \]
Using the change of base formula, we can rewrite the logarithms:
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