Prove that `p x^(q-r)+q x^(r-p)+r x^(p-q)> p+q+r ,where p, q, r` are distinct and `x!=1.`

Explanation

To prove that \( p x^{(q-r)} + q x^{(r-p)} + r x^{(p-q)} > p + q + r \) where \( p, q, r \) are distinct and \( x \neq 1 \), we can use the concept of weighted arithmetic mean and weighted geometric mean.

### Step-by-Step Solution:

1. **Identify the Terms**: We have three terms: \( p x^{(q-r)} \), \( q x^{(r-p)} \), and \( r x^{(p-q)} \). We need to show that their weighted sum is greater than the sum of the weights \( p + q + r \).

2. **Apply Weighted AM-GM Inequality**: According to the weighted AM-GM inequality, for non-negative weights \( p, q, r \) and corresponding terms \( x^{(q-r)}, x^{(r-p)}, x^{(p-q)} \):
\[
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