If the ratio (x + 1/x) : (x − 1/x) is 5 : 3 for a real number x ≠ 0, find the possible values of x.

Introduction / Context:
This problem tests algebraic manipulation involving expressions of the form x + 1/x and x − 1/x and a given ratio between them. Such questions are classic in algebra practice and aptitude tests. By converting the ratio relation into an equation and simplifying, we can derive a quadratic equation whose roots are the possible values of x.

Given Data / Assumptions:

• (x + 1/x) : (x − 1/x) = 5 : 3.

• x is a real number with x ≠ 0 (so that 1/x is defined).

• We must determine the real values of x consistent with the ratio.

Concept / Approach:
A ratio a : b = c : d implies a/b = c/d when all quantities are defined and b, d are non zero. Here we can write (x + 1/x) / (x − 1/x) = 5/3. Cross multiplication will give a linear equation in x and 1/x. Multiplying through by x to clear denominators turns this into a polynomial equation in x. Rearranging produces a quadratic equation which can be solved using standard techniques. The roots of this quadratic give the required values of x, provided they are real and non zero.

Step-by-Step Solution:
Step 1: Start from the ratio equation: (x + 1/x) / (x − 1/x) = 5 / 3. Step 2: Cross multiply: 3(x + 1/x) = 5(x − 1/x). Step 3: Expand both sides: 3x + 3/x = 5x − 5/x. Step 4: Bring like terms together: move terms with x to one side and terms with 1/x to the other side. Step 5: Subtract 3x from both sides: 3/x = 2x − 5/x. Step 6: Add 5/x to both sides: 3/x + 5/x = 2x, giving 8/x = 2x. Step 7: Multiply both sides by x (x ≠ 0): 8 = 2x^2. Step 8: Divide both sides by 2: x^2 = 4. Step 9: Taking square roots gives x = ±2.

Verification / Alternative check:
Check x = 2. Then x + 1/x = 2 + 1/2 = 2.5 and x − 1/x = 2 − 1/2 = 1.5. The ratio is 2.5 : 1.5 which simplifies to 5 : 3 after multiplying numerator and denominator by 2. For x = −2, we have x + 1/x = −2 − 1/2 = −2.5 and x − 1/x = −2 + 1/2 = −1.5. The ratio (−2.5) : (−1.5) again simplifies to 5 : 3 since the negatives cancel. Hence both x = 2 and x = −2 satisfy the given ratio.

Why Other Options Are Wrong:
Option +1 or −1 would give x − 1/x equal to zero, making the ratio undefined. Option +3 or −3 and option +4 or −4 do not satisfy the equation 8 = 2x^2 and produce different ratios. The option stating no real value is incorrect because we have found two real solutions. Only x = +2 or x = −2 produce the correct 5 : 3 ratio between x + 1/x and x − 1/x.

Common Pitfalls:
A common error is to forget that division by x − 1/x requires this quantity to be non zero, but this is automatically ensured at the correct solutions. Some students incorrectly cross multiply or mishandle the terms with 1/x, leading to a wrong equation. Others may forget to consider both positive and negative square roots when solving x^2 = 4. Writing each step carefully and explicitly resolving signs prevents these mistakes.

Final Answer:
The possible real values of x that satisfy the given ratio are +2 or −2.