Solve the linear equation −3[1 − (x/2)] + 5x/3 = 1/6 and determine the exact value of x.

Introduction / Context:
This equation combines brackets and fractional coefficients, which makes it a good test of basic algebra skills. You must distribute the negative sign correctly, handle the fractions carefully, and then collect like terms to solve for x. Such questions appear frequently in the simplification and algebra segments of aptitude exams.

Given Data / Assumptions:

• Equation: −3[1 − (x/2)] + 5x/3 = 1/6.
• x is a real number.
• Standard arithmetic and distributive laws apply.
• Fractions are defined (no division by zero occurs).

Concept / Approach:
The essential steps are to expand the bracket −3[1 − (x/2)], simplify the expression on the left, and then isolate x. One effective strategy is to clear denominators by multiplying the entire equation by the least common multiple of 2, 3 and 6. This converts the equation into one involving integers only, which is easier to solve without arithmetic errors.

Step-by-Step Solution:
Start with −3[1 − (x/2)] + 5x/3 = 1/6. Expand the bracket: 1 − (x/2). Multiply by −3: −3[1 − (x/2)] = −3 + (3x/2). So the equation becomes −3 + (3x/2) + 5x/3 = 1/6. Now multiply the entire equation by 6 (LCM of 2, 3 and 6): 6(−3) + 6(3x/2) + 6(5x/3) = 6(1/6). This gives −18 + 9x + 10x = 1. Combine like terms: −18 + 19x = 1. So 19x = 19, and x = 19/19 = 1.

Verification / Alternative check:
Substitute x = 1 back into the original equation. Inside the bracket we get 1 − (1/2) = 1/2. Thus −3[1 − (x/2)] = −3(1/2) = −3/2. Next, 5x/3 = 5(1)/3 = 5/3. So the left side becomes −3/2 + 5/3. With denominator 6, that is −9/6 + 10/6 = 1/6, which is exactly the right side. This confirms that x = 1 is the correct solution.

Why Other Options Are Wrong:
If x = 2 or x = −2, the left side evaluates to values different from 1/6. Similarly, x = −1 or x = 0 do not satisfy the original equation when substituted. Only x = 1 makes the entire equation balance with the right side of 1/6.

Common Pitfalls:
Common errors include distributing the negative sign incorrectly inside the bracket, forgetting to multiply every term when clearing denominators, or mishandling the addition of fractions. Another pitfall is to cancel terms prematurely, which can change the structure of the equation. Keeping track of each step with care helps prevent these mistakes.

Final Answer:
Therefore, the solution to the equation is x = 1.