Solve the linear equation x/3 − [5(7x/5 − 4/3)] / 2 = −x/6 and determine the exact value of x that satisfies this simplification problem.

Introduction / Context:
This algebra problem involves solving a linear equation that contains nested fractions. Questions of this type test your ability to simplify complex fractional expressions and handle operations in the correct order without losing track of signs or denominators.

Given Data / Assumptions:

• The equation is x/3 − [5(7x/5 − 4/3)] / 2 = −x/6.
• x is a real number.
• We want the unique value of x that satisfies this equation.

Concept / Approach:
The key idea is to simplify the nested expression step by step. First simplify the inner bracket, then multiply by 5, and then divide by 2. Once the left hand side is simplified, we will obtain a simple linear equation in x that we can solve using standard algebraic operations such as collecting like terms and isolating x.

Step-by-Step Solution:
Start with x/3 − [5(7x/5 − 4/3)] / 2 = −x/6. Simplify inside the bracket: 7x/5 − 4/3. Multiply by 5: 5(7x/5 − 4/3) = 7x − 20/3. Divide by 2: [5(7x/5 − 4/3)] / 2 = (7x − 20/3) / 2 = 7x/2 − 10/3. Now the equation becomes x/3 − (7x/2 − 10/3) = −x/6. Distribute the minus sign: x/3 − 7x/2 + 10/3 = −x/6. Combine the terms on the left using denominator 6: x/3 = 2x/6, 7x/2 = 21x/6, and 10/3 = 20/6. So left side is (2x/6 − 21x/6 + 20/6) = (−19x + 20) / 6. Thus (−19x + 20) / 6 = −x/6. Multiply both sides by 6: −19x + 20 = −x. Add 19x to both sides: 20 = 18x, so x = 20 / 18 = 10 / 9.
Verification / Alternative check:
Substitute x = 10 / 9 back into the original equation and simplify both sides. With careful fraction work, you will see that both sides become equal, confirming that x = 10 / 9 is correct.

Why Other Options Are Wrong:
Options b, c, and d produce different left hand side and right hand side values when substituted into the equation. Option e (0) clearly fails because the right hand side would be zero while the left hand side would not simplify to zero.

Common Pitfalls:
Common errors include incorrect distribution of the minus sign, mixing up denominators when combining fractions, and losing track of negative signs. Working carefully with a common denominator and checking each simplification step reduces these mistakes.

Final Answer:
The correct solution of the equation is x = 10 / 9.