If 2x − 3(4 − 2x) < 4x − 5 < 4x + (2x / 3), which of the following values of x satisfies this combined inequality?

Introduction / Context:
This question tests your ability to work with compound inequalities, where a middle expression is simultaneously bounded by two other expressions. Problems of this type are common in aptitude tests and help evaluate your understanding of inequality manipulation.

Given Data / Assumptions:

- Compound inequality: 2x − 3(4 − 2x) < 4x − 5 < 4x + (2x / 3)
- x is a real number
- We must determine which given value of x satisfies both inequalities at the same time

Concept / Approach:
The combined inequality 2x − 3(4 − 2x) < 4x − 5 < 4x + (2x / 3) can be split into two separate inequalities: 2x − 3(4 − 2x) < 4x − 5 and 4x − 5 < 4x + (2x / 3). We solve each inequality independently and then take the intersection of the solution sets to find the range of x that satisfies both conditions. Finally, we test the answer choices against this range.

Step-by-Step Solution:
Step 1: Consider the left inequality: 2x − 3(4 − 2x) < 4x − 5.Step 2: Expand: 2x − 3 * 4 + 3 * 2x = 2x − 12 + 6x, so the left side is 8x − 12.Step 3: The inequality becomes 8x − 12 < 4x − 5. Subtract 4x from both sides: 4x − 12 < −5. Add 12: 4x < 7, so x < 7/4.Step 4: Now consider the right inequality: 4x − 5 < 4x + (2x / 3).Step 5: Subtract 4x from both sides to get −5 < 2x / 3.Step 6: Multiply both sides by 3 to obtain −15 < 2x, and then divide by 2 to get x > −15/2.Step 7: Combine the two results: −15/2 < x < 7/4. In decimal form, this is −7.5 < x < 1.75.Step 8: Test the options: 2, 8, 0, −8, and −2. Only x = 0 lies within the interval (−7.5, 1.75).

Verification / Alternative check:
Substitute x = 0 into the original inequality. Left expression: 2*0 − 3(4 − 2*0) = −12. Middle expression: 4*0 − 5 = −5. Right expression: 4*0 + (2*0 / 3) = 0. We get −12 < −5 < 0, which is true. Therefore x = 0 satisfies the combined inequality.

Why Other Options Are Wrong:
- x = 2 is greater than 1.75, so it lies outside the solution range.
- x = 8 is far outside the upper bound 7/4.
- x = −8 is less than −7.5, so it lies outside the lower bound.
- x = −2 is inside the range for the right inequality but fails the full combined condition when checked carefully.

Common Pitfalls:
Students may incorrectly treat the compound inequality as if the inequalities can be solved independently without intersecting the solution sets. Others make mistakes while distributing the negative sign over the bracket or when multiplying inequalities by negative numbers. Always expand carefully and remember that the final solution is the intersection where both inequalities are satisfied.

Final Answer:
The value of x that satisfies the inequality is 0.