Solve the linear equation with fractional terms and find the exact value of x: 8x/3 + [7(5 - 2x/3)]/2 = 1/2.

Introduction / Context:
This algebra question involves solving a linear equation that contains fractions and brackets. You are required to simplify the expression carefully, clear the fractions, and then solve for x. Such problems are common in aptitude tests and algebra courses as they test attention to detail and comfort with fractional arithmetic and linear equations.

Given Data / Assumptions:

• Equation: 8x/3 + [7(5 − 2x/3)]/2 = 1/2.
• x is a real number.
• We must find the exact value of x that satisfies the equation.
• Standard arithmetic operations and fraction rules apply.

Concept / Approach:
First, simplify the bracketed term 7(5 − 2x/3) and then divide by 2 as indicated. Combine like terms, especially those involving x, to obtain a simpler linear equation. Clear fractions by finding a common denominator or by multiplying through by an appropriate number if needed. Finally, isolate x and solve. Throughout, careful handling of fractions and signs is essential to avoid mistakes.

Step-by-Step Solution:
1) Start with the equation: 8x/3 + [7(5 − 2x/3)]/2 = 1/2. 2) Simplify inside the bracket: 7(5 − 2x/3) = 7 * 5 − 7 * (2x/3) = 35 − 14x/3. 3) Divide by 2: [7(5 − 2x/3)]/2 = (35 − 14x/3) / 2 = 35/2 − (14x/3)/2 = 35/2 − 14x/6 = 35/2 − 7x/3. 4) Substitute back into the equation: 8x/3 + 35/2 − 7x/3 = 1/2. 5) Combine like x terms on the left: 8x/3 − 7x/3 = x/3, so the equation becomes x/3 + 35/2 = 1/2. 6) Subtract 35/2 from both sides: x/3 = 1/2 − 35/2 = (1 − 35)/2 = −34/2 = −17. 7) Multiply both sides by 3 to solve for x: x = 3 * (−17) = −51.

Verification / Alternative check:
Substitute x = −51 back into the original equation. Compute 8x/3 = 8(−51)/3 = −408/3 = −136. Then compute 5 − 2x/3 = 5 − 2(−51)/3 = 5 + 102/3 = 5 + 34 = 39. Next, 7(5 − 2x/3) = 7 * 39 = 273, and [7(5 − 2x/3)]/2 = 273/2. The left side becomes −136 + 273/2. Express −136 as −272/2, so −272/2 + 273/2 = 1/2, which matches the right side of the equation. This confirms that x = −51 is indeed the correct solution.

Why Other Options Are Wrong:
If you plug in −17, 17, or 51, the left-hand side will not simplify to 1/2. For example, with x = 17, the term 8x/3 becomes positive and large, and the bracketed term cannot compensate to produce 1/2. Since we have found a consistent and verified solution, the option 'None of these' is also incorrect. Among the given options, only −51 satisfies the equation.

Common Pitfalls:
Many students make errors when distributing over the bracket 7(5 − 2x/3) or when dividing by 2, especially in handling the x term in the fraction. Others mis subtract 35/2 on both sides or incorrectly simplify 1/2 − 35/2. Writing each fraction step explicitly and keeping a common denominator helps avoid these mistakes. Always verify the candidate solution by substitution to ensure accuracy.

Final Answer:
The solution of the equation is -51.