Solve the linear equation involving fractions carefully by clearing denominators.\nFind the value of x that satisfies: 14/3 + (1/2)(x - 7/3) = -2x/3.

Introduction / Context:
This question tests fraction-handling in linear equations. The core skill is to simplify bracketed fractional expressions and then eliminate denominators (by multiplying by a common multiple) to avoid arithmetic errors. Once simplified, it becomes a straightforward one-variable equation.

Given Data / Assumptions:

• Equation: 14/3 + (1/2)(x - 7/3) = -2x/3
• x is a real number.
• Goal: solve for x exactly.

Concept / Approach:
Distribute (1/2) across the bracket, convert all terms to a common denominator, and then multiply both sides by the LCM of denominators (here 6) to convert to integers. This prevents mistakes from repeated fraction operations.

Step-by-Step Solution:
Start: 14/3 + (1/2)(x - 7/3) = -2x/3 Distribute 1/2: (1/2)(x - 7/3) = x/2 - 7/6 Rewrite 14/3 as /6: 14/3 = 28/6 Left side becomes: 28/6 + x/2 - 7/6 = (21/6) + x/2 So: 7/2 + x/2 = -2x/3 Multiply both sides by 6: 6*(7/2 + x/2) = 6*(-2x/3) Left: 6*(7/2) = 21 and 6*(x/2) = 3x, so 21 + 3x Right: 6*(-2x/3) = -4x Equation: 21 + 3x = -4x Add 4x: 21 + 7x = 0 So 7x = -21 => x = -3

Verification / Alternative check:
Substitute x = -3 into the original: (1/2)(-3 - 7/3) = (1/2)(-16/3) = -8/3. Then 14/3 - 8/3 = 6/3 = 2, and RHS is -2*(-3)/3 = 2. Both sides match.

Why Other Options Are Wrong:
-6 and 6 come from sign or distribution errors.
0 and 3 typically result from incorrect handling of the negative RHS or combining fractions incorrectly.

Common Pitfalls:
Forgetting to distribute 1/2 to both terms, or using an incorrect LCM when clearing denominators. Another frequent mistake is dropping the negative sign in -2x/3.

Final Answer:
x = -3