In basic algebraic simplification, evaluate the expression a/(a - b) + b/(b - a) for a ≠ b, simplify it completely, and then choose the correct value from the given options.

Introduction / Context:
This problem checks your understanding of algebraic fractions and sign manipulation. The goal is to simplify the expression a/(a - b) + b/(b - a) under the assumption that a is not equal to b, so that denominators are non zero. You must carefully handle the fact that the two denominators are negatives of one another and then pick the correct simplified result.

Given Data / Assumptions:

• The expression is a/(a - b) + b/(b - a).
• a and b are real numbers with a ≠ b.
• Division by zero is not allowed, so denominators are valid.
• We are expected to simplify to the simplest possible algebraic form.

Concept / Approach:
Notice that b - a is the negative of a - b, that is, b - a = -(a - b). This makes the second fraction easy to rewrite with a common denominator. Once the denominators match, the numerators can be combined directly. Simplifying step by step reveals that the final answer is a constant rather than an expression involving a and b.

Step-by-Step Solution:
Start with the expression: a/(a - b) + b/(b - a). Recognise that b - a = -(a - b), so b/(b - a) = b/(-(a - b)) = -b/(a - b). Rewrite the sum using a common denominator: a/(a - b) - b/(a - b). Combine numerators: (a - b)/(a - b) = 1, since a ≠ b ensures the denominator is non zero.
Verification / Alternative check:
You can verify the result by substituting simple numeric values for a and b, such as a = 5 and b = 3. The left side becomes 5/(2) + 3/(-2) = 5/2 - 3/2 = 2/2 = 1, which matches the simplified expression. Trying several pairs confirms that the result does not depend on the particular values, only on the condition that a ≠ b.

Why Other Options Are Wrong:
Option b ( -1 ) would require the numerator to simplify to -(a - b), which is not the case. Option c ( 2ab ) incorrectly multiplies variables instead of simplifying fractions. Options d and e suggest that the expression remains a rational function in a and b, but the algebra shows it simplifies to a constant. Therefore they are not correct.

Common Pitfalls:
Learners often forget that b - a is the negative of a - b and may mistakenly add the fractions without adjusting the sign, leading to incorrect numerators. Another pitfall is attempting to cross multiply unnecessarily instead of first recognising the simple sign relationship between the denominators.

Final Answer:
The simplified value of a/(a - b) + b/(b - a) is 1 for all a and b with a ≠ b.