For a non-zero real number x and real numbers p and q, consider the expression: 1/x^(p − q + 1) + 1/x^(q − p + 1). Without any additional relationship between p and q, which statement best describes the value of this expression?

Introduction / Context:
This question tests whether an algebraic expression can be uniquely evaluated or simplified to a single fixed value when key information is missing. Many exponent expressions look symmetric, but symmetry alone does not force a constant result unless we know something like p = q, p − q = 1, or a specific value of x.

Given Data / Assumptions:

• x is a non-zero real number • p and q are real numbers with no extra constraints given • Expression: 1/x^(p − q + 1) + 1/x^(q − p + 1)

Concept / Approach:
Rewrite each term using negative exponents to see the structure: 1/x^(p − q + 1) = x^(−(p − q + 1)) = x^(q − p − 1) 1/x^(q − p + 1) = x^(p − q − 1) So the expression becomes: x^(q − p − 1) + x^(p − q − 1). This depends on both (p − q) and x. Without knowing either a relation between p and q or a numerical value of x, it can take many different values.

Step-by-Step Solution:
1) Convert to exponent form: E = x^(q − p − 1) + x^(p − q − 1) 2) Observe: if p = q, then E = x^(−1) + x^(−1) = 2/x 3) If p − q = 1, then E = x^(−2) + x^(0) = 1/x^2 + 1 4) These examples show E changes with different p, q choices.

Verification / Alternative check:
Take x = 2: If p = q, E = 2/2 = 1. If p − q = 1, E = 1/4 + 1 = 1.25. Same x, different (p, q), different value. So it is not uniquely determined.

Why Other Options Are Wrong:
• 0, 1, x, x^2: these claim a fixed value, but the expression varies with p and q.

Common Pitfalls:
• Assuming symmetry forces a constant result. • Treating p and q as integers or assuming p = q without it being stated.

Final Answer:
Cannot be determined uniquely from the information given