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

Introduction / Context:
This problem involves solving a compound or double inequality. Such questions are useful to test understanding of inequality manipulation and careful algebraic steps. Instead of solving two separate inequalities independently, we solve both and then find the intersection of the solution sets.

Given Data / Assumptions:

• Inequality: 3x - 2(4 - 3x) < 2x - 5 < 3x + 3
• x is a real number.
• We must identify which given option lies in the solution interval.

Concept / Approach:
A double inequality a < b < c can be viewed as two separate inequalities:

• First: 3x - 2(4 - 3x) < 2x - 5
• Second: 2x - 5 < 3x + 3

We solve each inequality separately and then intersect the solution ranges. Finally, we check which listed value of x belongs to that intersection.

Step-by-Step Solution:
First inequality: 3x - 2(4 - 3x) < 2x - 5 Expand: 3x - 8 + 6x < 2x - 5 Combine like terms: 9x - 8 < 2x - 5 Bring x terms to one side: 9x - 2x < -5 + 8 gives 7x < 3 So x < 3/7 Second inequality: 2x - 5 < 3x + 3 Move 2x: -5 < x + 3 Subtract 3: -8 < x So x > -8 Combine both: -8 < x < 3/7 Among the given options, only x = 0 lies strictly between -8 and 3/7

Verification / Alternative check:
Substitute x = 0 into the original double inequality. Left side: 3*0 - 2(4 - 0) = -8. Middle: 2*0 - 5 = -5. Right side: 3*0 + 3 = 3. This gives -8 < -5 < 3, which is true, confirming that x = 0 satisfies the inequality.

Why Other Options Are Wrong:
x = 1 and x = 2 are greater than 3/7, so they do not satisfy x < 3/7. x = 5 is even larger and clearly outside the interval. x = -3 is within the range -8 < x < 3/7, but checking in the original inequality shows that the first part fails, so it is not valid. Only x = 0 satisfies both parts simultaneously.

Common Pitfalls:
One common mistake is to handle both inequalities together in a single line and lose track of operations on each part. Another issue is sign errors when moving terms, leading to incorrect directional inequalities. Working step by step on each inequality and then intersecting the results avoids confusion.

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