In this aptitude (simplification and inequalities) question, consider the compound inequality: 6x + 2(6 - x) > 2x - 2 < 5x/2 - 3x/4. Interpret this as two separate inequalities involving the central expression 2x - 2, solve them, and then decide which value of x among the options satisfies the entire inequality.

Introduction / Context:
This question presents a chain inequality, where one expression is bounded by two others. Such inequalities require careful interpretation and algebraic manipulation. The task is to split the chain into separate inequalities, solve each one, and then find the intersection of the resulting ranges to determine which option for x satisfies the full condition.

Given Data / Assumptions:
We are given 6x + 2(6 - x) > 2x - 2 < 5x/2 - 3x/4.
This means:
First inequality: 6x + 2(6 - x) > 2x - 2.
Second inequality: 2x - 2 < 5x/2 - 3x/4.

Concept / Approach:
Solve each inequality separately, simplifying expressions step by step. After we have the solution set for each inequality, we take their intersection because x must satisfy both simultaneously. Finally, we test which given option lies within this common solution interval. Paying attention to arithmetic with fractions is important in the second inequality.

Step-by-Step Solution:
First inequality: 6x + 2(6 - x) > 2x - 2. Expand: 6x + 12 - 2x > 2x - 2. Simplify: 4x + 12 > 2x - 2. Subtract 2x: 2x + 12 > -2. Subtract 12: 2x > -14, so x > -7. Second inequality: 2x - 2 < 5x/2 - 3x/4. Compute the right side: 5x/2 - 3x/4 = (10x/4 - 3x/4) = 7x/4. So we have 2x - 2 < 7x/4. Multiply through by 4 to clear the denominator: 8x - 8 < 7x. Subtract 7x: x - 8 < 0, so x < 8. Intersection of the two ranges is -7 < x < 8.

Verification / Alternative check:
Now test the options: 5, 9, -8, -9, and 8. Only numbers strictly between -7 and 8 satisfy both inequalities. The number 5 lies in this interval, whereas 9, 8, -8, and -9 do not. We can also verify directly by substituting x = 5 into the original chain. For x = 5, the left expression is 6*5 + 2(6 - 5) = 30 + 2 = 32. The central expression is 2*5 - 2 = 8. The right expression is 5*5/2 - 3*5/4 = 25/2 - 15/4 = 50/4 - 15/4 = 35/4, which is 8.75. We indeed have 32 > 8 and 8 < 8.75, so the chain holds.

Why Other Options Are Wrong:
Option 9 and 8 violate the condition x < 8 from the second inequality. Option -8 and -9 violate the condition x > -7 from the first inequality. Only 5 lies in the intersection of the solution sets and also satisfies the original inequality when checked directly.

Common Pitfalls:
Students sometimes misinterpret the compound inequality, treating it as three separate unrelated comparisons instead of two connected inequalities. Another pitfall is mishandling fractional coefficients or forgetting to multiply both sides by the same factor when clearing denominators. Keeping the work organised and carefully transforming one step at a time avoids these errors.

Final Answer:
The value of x that satisfies the entire compound inequality is 5.