If 5x - 3(4 - x) < 4x - 4 < 4x + 2x/3, then which value of x from the options satisfies this compound inequality?

Introduction / Context:
This question involves a compound inequality made of two separate inequalities joined by a common middle expression. The task is to find values of x that satisfy both inequalities simultaneously. Such questions test your skill in solving linear inequalities, simplifying expressions, and understanding solution intervals.

Given Data / Assumptions:

• Inequality 1: 5x - 3(4 - x) < 4x - 4.
• Inequality 2: 4x - 4 < 4x + 2x/3.
• We must choose x from the options that satisfies both inequalities.

Concept / Approach:
We solve each inequality separately. The first one simplifies to a standard linear inequality in x. The second inequality, with the same left side 4x - 4, simplifies quickly. After obtaining the solution intervals from both inequalities, we intersect them to get the common range of admissible x values. Finally, we pick the option that lies in this common interval.

Step-by-Step Solution:

Verification / Alternative check:
Test x = 1 directly in both inequalities. For the first: 5*1 - 3(4 - 1) = 5 - 9 = -4 and 4*1 - 4 = 0. Since -4 < 0, the first inequality holds. For the second: 4*1 - 4 = 0 and 4*1 + 2*1/3 = 4 + 2/3 = 4.666..., and 0 < 4.666..., so the second inequality also holds. This confirms that x = 1 satisfies the compound inequality.

Why Other Options Are Wrong:

Common Pitfalls:
Errors often occur when expanding 3(4 - x), especially with sign changes. Another common mistake is to mix up directions of inequalities when dividing by a negative number. Here, we only divide by positive numbers, so the inequality signs do not flip. Writing each step clearly helps avoid these issues.

Final Answer:
1