Solve the equation x/2 − [4(15/2 − x/3)]/3 = −x/18 and find the value of x.

Introduction / Context:
This linear equation contains nested fractions and requires careful handling of arithmetic operations. Being able to simplify such expressions accurately is important in algebra and in many quantitative aptitude questions where fractional coefficients appear regularly.

Given Data / Assumptions:

• The equation is x/2 − [4(15/2 − x/3)]/3 = −x/18.
• x is a real number.
• We need to isolate x and find its value.

Concept / Approach:
We handle the bracket 15/2 − x/3 first, multiply by 4, then divide by 3, and finally simplify the equation. To avoid errors with denominators, it is efficient to multiply through by the least common multiple of all denominators so that the resulting equation has integer coefficients. Afterwards, standard linear solving steps are applied.

Step-by-Step Solution:
Start with x/2 − [4(15/2 − x/3)]/3 = −x/18. First simplify the inner expression: 15/2 − x/3. Multiply by 4: 4(15/2 − x/3) = 4 * 15/2 − 4 * x/3 = 30 − 4x/3. Now divide by 3: [4(15/2 − x/3)]/3 = (30 − 4x/3)/3. Rewrite as 30/3 − (4x/3)/3 = 10 − 4x/9. So the equation becomes x/2 − (10 − 4x/9) = −x/18. Distribute the minus sign: x/2 − 10 + 4x/9 = −x/18. Take the least common multiple of denominators 2, 9, and 18, which is 18, and multiply through. 18 * (x/2) = 9x, 18 * (−10) = −180, 18 * (4x/9) = 8x, and 18 * (−x/18) = −x. Thus 9x − 180 + 8x = −x. Combine like terms: 17x − 180 = −x. Add x to both sides: 18x − 180 = 0. Add 180: 18x = 180, so x = 180 / 18 = 10.

Verification / Alternative check:
Substitute x = 10 back into the original equation. The left side becomes 10/2 − [4(15/2 − 10/3)]/3. Compute 10/2 = 5. Inside the bracket, 15/2 − 10/3 = 45/6 − 20/6 = 25/6. Multiplying by 4 gives 100/6 = 50/3. Dividing by 3 gives 50/9. So the left side is 5 − 50/9 = (45/9 − 50/9) = −5/9. The right side is −x/18 = −10/18 = −5/9. Both sides match, confirming that x = 10 is correct.

Why Other Options Are Wrong:
Values such as −10 or 9/8 result from sign errors or mismanagement of denominators during simplification. The value −9/8 comes from incorrect inversion of a fraction, and 15 arises from miscalculating when clearing denominators. None of these, when substituted back, satisfy the original equation.

Common Pitfalls:
Common mistakes include dropping a minus sign in front of the bracket, incorrectly distributing 4 over the fraction, or not multiplying every term by the least common multiple when clearing denominators. Taking the time to simplify step by step and checking the solution in the original equation helps avoid such errors.

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