If 5x + 4(1 - x) > 3x - 4 > (5x/3 - x/3), which of the following values can x take?

Introduction / Context:
This problem involves solving a compound or double inequality. We have an expression that must satisfy two inequalities at the same time. Such questions are typical in aptitude tests because they combine algebraic manipulation and careful reasoning about ranges of values for x.

Given Data / Assumptions:

• Inequality 1: 5x + 4(1 - x) > 3x - 4
• Inequality 2: 3x - 4 > (5x/3 - x/3)
• We must find which option value of x satisfies both inequalities simultaneously.

Concept / Approach:
A double inequality can be treated as two separate inequalities that must both hold. We solve each inequality independently to find the range of x that satisfies it. Finally, we take the intersection of the two solution sets. Only values that lie in this intersection are valid, and we then check which option matches this range.

Step-by-Step Solution:
Step 1: Simplify the left expression in Inequality 1: 5x + 4(1 - x) = 5x + 4 - 4x = x + 4. Step 2: Inequality 1 becomes x + 4 > 3x - 4. Step 3: Rearrange to get -2x + 8 > 0, so -2x > -8 and x < 4. Step 4: Simplify the right side of Inequality 2: 5x/3 - x/3 = 4x/3. Step 5: Inequality 2 becomes 3x - 4 > 4x/3. Step 6: Multiply through by 3 to clear the denominator: 9x - 12 > 4x, so 5x > 12 and x > 12/5. Step 7: Combine the two results: x must satisfy 12/5 < x < 4, that is 2.4 < x < 4. Step 8: From the given options, only x = 3 lies in the interval (2.4, 4).

Verification / Alternative check:
Substitute x = 3 back into the original inequalities. For x = 3, the left side is 5*3 + 4(1 - 3) = 15 - 8 = 7 and the middle part is 3*3 - 4 = 5, so 7 > 5 is true. The right comparison is 3*3 - 4 = 5 and 5x/3 - x/3 = 4x/3 = 4, so 5 > 4 is also true. Hence x = 3 satisfies both parts.

Why Other Options Are Wrong:

• Option A: x = 2 is less than 12/5 and fails the second inequality.
• Option B: x = 1 clearly fails both inequalities.
• Option D: x = -2 fails both inequalities by a large margin.
• Option E: x = 0 is outside the derived interval and does not satisfy the original conditions.

Common Pitfalls:
Errors often occur when simplifying expressions like 4(1 - x) or when handling fractions such as 5x/3 - x/3. Some students also forget that both inequalities must hold at the same time, so they may pick a value that satisfies only one of them. Keeping track of inequality directions and solving step by step is essential.

Final Answer:
Therefore, the value of x that satisfies both inequalities is 3.