In this aptitude (simplification and inequalities) question, solve the compound inequality: 4x - 5(2x - 1) > 2x + 3 > 2 - 3x, and determine which of the given values of x lies in the common solution range.

Introduction / Context:
This question involves solving a double inequality, where a central expression is bounded by two other expressions. These types of problems test algebraic manipulation as well as understanding of how to combine solution sets from multiple inequalities. The final step is to check which option for x lies in the intersection of these ranges.

Given Data / Assumptions:
We are given 4x - 5(2x - 1) > 2x + 3 > 2 - 3x.
This is equivalent to two separate inequalities that both must hold:
First: 4x - 5(2x - 1) > 2x + 3.
Second: 2x + 3 > 2 - 3x.

Concept / Approach:
We solve each inequality separately to find a range for x. Then we take the intersection of the two ranges, since x must satisfy both conditions simultaneously. After obtaining the final range, we simply check which of the given discrete options falls inside that interval.

Step-by-Step Solution:
First inequality: 4x - 5(2x - 1) > 2x + 3. Expand: 4x - 10x + 5 > 2x + 3. Simplify: -6x + 5 > 2x + 3. Bring x terms together: -6x - 2x > 3 - 5, so -8x > -2. Divide by -8 (reverse the inequality): x < 1/4. Second inequality: 2x + 3 > 2 - 3x. Add 3x to both sides: 5x + 3 > 2. Subtract 3: 5x > -1, so x > -1/5. Intersection of ranges is -1/5 < x < 1/4.

Verification / Alternative check:
Now test each option. For x = 0, both inequalities should hold. Substitute into first: 4*0 - 5(0 - 1) = 5, and 2*0 + 3 = 3, so 5 > 3 is true. For the second inequality, 2*0 + 3 = 3 and 2 - 3*0 = 2, so 3 > 2 is true. Hence x = 0 satisfies both. Other options either exceed 1/4 or are less than or equal to -1/5, so they do not lie in the interval.

Why Other Options Are Wrong:
Option 1, 2, and 3 are all greater than 1/4, violating the first inequality. Option -1 is less than -1/5, violating the second inequality. Only 0 lies strictly between -1/5 and 1/4, satisfying both inequalities simultaneously.

Common Pitfalls:
Students sometimes forget to reverse the inequality sign when dividing by a negative number. Another frequent error is to mix up which expressions are being compared, especially in a chain of inequalities. To avoid confusion, always break a compound inequality into two separate parts, solve each carefully, and then intersect the solution sets.

Final Answer:
The value of x among the options that satisfies both inequalities is 0.