Difficulty: Medium
Correct Answer: ± 8
Explanation:
Introduction / Context:The original database text was garbled. Using the Recovery-First Policy, we interpret it as a classic rational-expression identity: (x + 4)/(x − 4) + (x − 4)/(x + 4) = 10/3. This kind of expression simplifies via a common denominator, producing an equation only in x^2. We then solve for x and check domain restrictions (x ≠ ±4).
Given Data / Assumptions:
Concept / Approach:Use the identity (A/B) + (B/A) = (A^2 + B^2)/(AB). Here, A = x + 4 and B = x − 4. This yields a rational equation in x that simplifies to a quadratic in x^2. Solve and select the real roots allowed by the domain.
Step-by-Step Solution:
(x + 4)/(x − 4) + (x − 4)/(x + 4) = [(x + 4)^2 + (x − 4)^2]/(x^2 − 16) Numerator = (x^2 + 8x + 16) + (x^2 − 8x + 16) = 2x^2 + 32 So (2x^2 + 32)/(x^2 − 16) = 10/3 Cross-multiply: 3(2x^2 + 32) = 10(x^2 − 16) ⇒ 6x^2 + 96 = 10x^2 − 160 Rearrange: 4x^2 = 256 ⇒ x^2 = 64 ⇒ x = ±8Verification / Alternative check:Denominator check: x ≠ ±4, so ±8 are valid. Substitute x = 8 (or −8) to confirm the left-hand side equals 10/3 numerically.
Why Other Options Are Wrong:±4 are excluded (division by zero). ±6 and 2 ± √3 do not satisfy the equation when substituted. “No real root” is false since ±8 work.
Common Pitfalls:Dropping the domain restriction, or algebra errors when simplifying the rational sum can lead to spurious answers.
Final Answer:± 8
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