Solve the linear equation 7x - [3(2x - 3)]/2 = 1/2 by clearing the fraction and simplifying step by step, then choose the correct value of x from the options.

Introduction / Context:
This simplification problem presents a linear equation involving a bracket and a fraction. Questions like this are used in aptitude exams to test basic algebraic manipulation skills, such as expanding brackets, working with fractions, and collecting like terms properly. Once the equation is simplified, you can solve for x and match it with one of the given answer choices.

Given Data / Assumptions:

• The equation is 7x - [3(2x - 3)]/2 = 1/2.
• x is a real number.
• Fractions should be handled exactly without rounding.
• Multiplication and distribution inside brackets follow normal algebraic rules.

Concept / Approach:
The main idea is to remove the bracket by distributing the factor 3, then handle the division by 2. After that, multiply the whole equation by 2 to clear the denominator and obtain a simpler linear equation in x. Using standard algebraic steps such as combining like terms and isolating x will lead you to the correct solution efficiently.

Step-by-Step Solution:
Start with 7x - [3(2x - 3)]/2 = 1/2. First expand the bracket: 3(2x - 3) = 6x - 9. Substitute: 7x - (6x - 9)/2 = 1/2. Multiply the entire equation by 2 to clear the denominator: 2 * 7x - (6x - 9) = 2 * (1/2). This gives 14x - (6x - 9) = 1. Distribute the minus sign: 14x - 6x + 9 = 1. Combine like terms: 8x + 9 = 1. Solve for x: 8x = 1 - 9 = -8, so x = -8 / 8 = -1.
Verification / Alternative check:
Substitute x = -1 back into the original equation to verify. Compute 2x - 3 = 2(-1) - 3 = -2 - 3 = -5, so 3(2x - 3) = 3 * (-5) = -15. Then [3(2x - 3)]/2 = -15/2. The left hand side becomes 7(-1) - (-15/2) = -7 + 15/2 = (-14/2 + 15/2) = 1/2, which matches the right hand side. This confirms that x = -1 is the correct solution.

Why Other Options Are Wrong:
Option b (1), option c (3), and option d (-3) all give left hand side values that are not equal to 1/2 when substituted into the equation. Option e (0) corresponds to ignoring some terms or incorrectly simplifying the expression. Only x = -1 satisfies the equation exactly after full substitution.

Common Pitfalls:
Common errors include failing to distribute the minus sign correctly when removing the bracket, such as writing 14x - 6x - 9 instead of 14x - 6x + 9. Another pitfall is to forget to multiply every term, including the right hand side, when clearing denominators. Keeping track of all terms while multiplying and simplifying step by step is essential to avoid these mistakes.

Final Answer:
The correct solution of the equation 7x - [3(2x - 3)]/2 = 1/2 is x = -1.