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If x + 1/x = c + 1/c (with x ≠ 0 and c ≠ 0), what are the possible values of x in terms of c? Choose the correct set of solutions.

Difficulty: Medium

Correct Answer: x = c or x = 1/c

Explanation:


Introduction / Context:
This identity-based question tests equation manipulation involving a variable and its reciprocal. A common trick is to bring everything to one side and factor using symmetry. The solutions typically come in reciprocal pairs.


Given Data / Assumptions:

  • x ≠ 0 and c ≠ 0
  • Equation: x + 1/x = c + 1/c


Concept / Approach:
Rewrite the equation and clear denominators. After simplification, it becomes a quadratic in x whose roots are c and 1/c. Another way is to show (x - c)(x - 1/c) = 0.


Step-by-Step Solution:

Start: x + 1/x = c + 1/c. Bring terms together: (x - c) + (1/x - 1/c) = 0. Combine the reciprocal difference: 1/x - 1/c = (c - x)/(xc). So (x - c) + (c - x)/(xc) = 0. Note c - x = -(x - c), so equation becomes (x - c) - (x - c)/(xc) = 0. Factor (x - c): (x - c) * (1 - 1/(xc)) = 0. Therefore either x - c = 0 giving x = c, or 1 - 1/(xc) = 0 giving xc = 1, so x = 1/c.


Verification / Alternative check:
If x = c, LHS equals RHS immediately. If x = 1/c, then x + 1/x = 1/c + c, which is the same as c + 1/c, so it also satisfies the equation.


Why Other Options Are Wrong:

x = c^2 or x = 2c does not preserve the reciprocal symmetry. x = 0 is invalid because 1/x is undefined. x = -c or -1/c would give -(c + 1/c), not c + 1/c.


Common Pitfalls:
Forgetting the non-zero condition, or incorrectly combining 1/x - 1/c. Also, many people miss the reciprocal solution x = 1/c.


Final Answer:
x = c or x = 1/c

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