Solve the compound inequality 4(2x - 4) - 2 > 3x - 1 ≥ 4x - 7 and determine which of the following values x can take so that the entire chained inequality holds true.

Introduction / Context:
This question involves a chained inequality with three algebraic expressions. The notation 4(2x - 4) - 2 > 3x - 1 ≥ 4x - 7 means that both inequalities must hold at the same time. Your goal is to solve this compound inequality and then see which given value of x lies in the resulting solution set. Such chained inequalities appear in aptitude tests to check comfort with solving and combining multiple linear inequalities.

Given Data / Assumptions:

• First part: 4(2x - 4) - 2 > 3x - 1.
• Second part: 3x - 1 ≥ 4x - 7.
• Both must be satisfied simultaneously.
• We must choose which option value of x satisfies the compound inequality.

Concept / Approach:
We break the chained inequality into two separate inequalities and solve each one step by step. Each part produces an interval of x values. The final solution set is the intersection (overlap) of these intervals because both conditions must hold at once. After finding the intersection, we check which option lies inside this combined range. Only one option is chosen so that exactly one is correct.

Step-by-Step Solution:
First inequality: 4(2x - 4) - 2 > 3x - 1. Expand: 4 * 2x = 8x and 4 * (-4) = -16, so 8x - 16 - 2 > 3x - 1. Simplify left side: 8x - 18 > 3x - 1. Subtract 3x from both sides: 5x - 18 > -1. Add 18 to both sides: 5x > 17. Divide by 5 (positive): x > 17/5 = 3.4. Second inequality: 3x - 1 ≥ 4x - 7. Subtract 3x from both sides: -1 ≥ x - 7. Add 7 to both sides: 6 ≥ x, which is x ≤ 6. Now combine: x > 3.4 and x ≤ 6. So the solution set is 3.4 < x ≤ 6. Among the options {1, 3, 4, 7, 10}, only x = 4 lies strictly above 3.4 and at or below 6.

Verification / Alternative check:
Check x = 4 in both inequalities. First part: 4(2 * 4 - 4) - 2 = 4(8 - 4) - 2 = 4 * 4 - 2 = 16 - 2 = 14. Right side is 3x - 1 = 12 - 1 = 11. Since 14 > 11, the first inequality holds. Second part: 3x - 1 = 11 and 4x - 7 = 16 - 7 = 9, so 11 ≥ 9, which also holds. Thus x = 4 satisfies the chain.

Why Other Options Are Wrong:
x = 1 and x = 3 are not greater than 3.4, so they fail the first inequality. x = 7 and x = 10 are greater than 6, so they fail the second inequality. Hence none of these values belong to the solution set 3.4 < x ≤ 6.

Common Pitfalls:
Some learners try to manipulate the entire chain in one go and lose track of signs, or they forget that the inequalities describe an interval that must be intersected, not merely united. Another mistake is to ignore the strictness of the inequality and accidentally include boundary points that should be excluded. Always solve each piece separately and clearly write the final interval before testing options.

Final Answer:
The value of x that satisfies the full compound inequality is 4.