Solve the linear equation [2(4x/5 − 3/4)] / 3 − 5/3 = −1/6 and determine the exact value of x that satisfies this fractional simplification problem.

Introduction / Context:
This question reinforces skills in solving linear equations involving multiple fractions and brackets. Such problems appear frequently in aptitude tests because they check attention to detail, especially with signs and denominators.

Given Data / Assumptions:

• The equation is [2(4x/5 − 3/4)] / 3 − 5/3 = −1/6.
• x is a real number.
• We must find the value of x that satisfies this equation.

Concept / Approach:
The strategy is to simplify the expression step by step. First simplify inside the brackets, then multiply by 2, then divide by 3. After that, we combine all fractions on one side and obtain a simple linear equation in x. Solving that equation gives the required value of x.

Step-by-Step Solution:
Start with [2(4x/5 − 3/4)] / 3 − 5/3 = −1/6. Simplify inside: 4x/5 − 3/4. Multiply by 2: 2(4x/5 − 3/4) = 8x/5 − 3/2. Now divide by 3: [2(4x/5 − 3/4)] / 3 = (8x/5 − 3/2) / 3 = 8x/15 − 1/2. So the equation becomes 8x/15 − 1/2 − 5/3 = −1/6. Combine constants −1/2 and −5/3 using common denominator 6: −1/2 = −3/6, −5/3 = −10/6, sum is −13/6. Thus 8x/15 − 13/6 = −1/6. Add 13/6 to both sides: 8x/15 = −1/6 + 13/6 = 12/6 = 2. So 8x/15 = 2, giving x = 2 * 15 / 8 = 30 / 8 = 15 / 4.
Verification / Alternative check:
Substitute x = 15 / 4 back into the original equation and simplify numerically. You find that both sides reduce to −1/6, confirming that 15 / 4 is indeed the correct solution.

Why Other Options Are Wrong:
Options a, b, and c produce mismatched left and right hand sides when substituted. Option e (0) clearly fails because the left side becomes negative while the right side is −1/6. Only x = 15 / 4 balances the equation exactly.

Common Pitfalls:
Learners frequently make mistakes when dealing with several fractions, especially in combining terms and handling negative signs. Sometimes they drop the division by 3 or incorrectly expand the numerator. Proceeding slowly and using a clear common denominator step helps avoid such errors.

Final Answer:
The solution of the linear equation is x = 15 / 4.