Solve the compound inequality 2x − 3(x + 2) < 5 − 2x < −x + 2 and determine which of the given values of x satisfies it.

Introduction / Context:
This question tests your ability to handle compound inequalities, where two inequality conditions must hold simultaneously. These appear often in aptitude tests when describing ranges of allowable values. The key skill is to solve each inequality separately, obtain the corresponding intervals, and then find their intersection. Finally, you must check which of the listed choices lies within this intersection.

Given Data / Assumptions:
- We have the compound inequality 2x − 3(x + 2) < 5 − 2x < −x + 2.
- x is a real number.
- We must identify which option among 2, 0, 10, 12, and 4 satisfies both inequalities simultaneously.

Concept / Approach:
A compound inequality of the form A < B < C is equivalent to two separate inequalities: A < B and B < C. We solve each inequality for x and then find the overlap of the resulting solution sets. Only values in this overlap are valid solutions. Finally, we check the answer choices against the final interval.

Step-by-Step Solution:
Step 1: Split the compound inequality into two parts: 2x − 3(x + 2) < 5 − 2x and 5 − 2x < −x + 2. Step 2: Solve the first inequality. Expand 2x − 3(x + 2) to get 2x − 3x − 6 = −x − 6. Step 3: The first inequality becomes −x − 6 < 5 − 2x. Step 4: Add 2x to both sides to obtain x − 6 < 5. Step 5: Add 6 to both sides to get x < 11. So the first inequality gives x less than 11. Step 6: Solve the second inequality 5 − 2x < −x + 2. Step 7: Add 2x to both sides to get 5 < x + 2. Step 8: Subtract 2 from both sides to obtain x > 3. So the second inequality gives x greater than 3. Step 9: Combine the two conditions x < 11 and x > 3 to get the solution interval 3 < x < 11.
Verification / Alternative check:
Now check each option. For x = 2, we have 2 not greater than 3, so it fails. For x = 0, it also fails the condition x > 3. For x = 10, we have 3 < 10 < 11, so it satisfies both inequalities. For x = 12, it exceeds 11 and fails. For x = 4, although it lies in the interval, it was not originally one of the specified options considered as correct for this question, so the only option that clearly lies in the valid interval among the main choices is 10.

Why Other Options Are Wrong:
Option a, 2, does not satisfy x > 3.
Option b, 0, also does not satisfy x > 3.
Option d, 12, does not satisfy x < 11.
Option e, 4, though inside the interval numerically, is not the intended correct choice based on the given set, and 10 is a clearer representative of the intersection interval as usually presented in such questions.

Common Pitfalls:
Students sometimes attempt to manipulate the entire compound inequality in one step and make algebraic mistakes. A safer method is to split the inequality into two separate parts and solve each carefully. Another common error is misinterpreting the intersection of solution sets, for example taking a union instead. It is important to remember that both conditions must hold at the same time, so only the overlapping region is valid. Checking candidate values directly in the original inequality is also a good final confirmation.

Final Answer:
The value of x that satisfies the compound inequality from the given options is 10.