Solve the compound system of linear inequalities in one variable: 2x − 2(3 + 4x) < −1 − 2x and −1 − 2x > (−5/3) − (x/3). Among the given options, which value of x is the greatest integer that satisfies both inequalities?

Introduction / Context:
This problem involves solving a system of two linear inequalities in a single variable x. Instead of just solving one inequality, you must find the intersection of the solution sets of both inequalities and then decide which of the given options fits an additional condition. Here, you are asked specifically for the greatest integer value among the options that satisfies both inequalities. This tests algebraic manipulation, understanding of inequalities, and reasoning about integer solutions.

Given Data / Assumptions:

• The first inequality is 2x − 2(3 + 4x) < −1 − 2x.
• The second inequality is −1 − 2x > (−5/3) − (x/3).
• x is a real variable, but the options represent integer candidates.
• We must find which option is the greatest integer that satisfies both inequalities simultaneously.

Concept / Approach:
We solve each inequality separately and express the solution sets as intervals on the real number line. Then we intersect these intervals to obtain the combined solution region. Finally, we look at the given answer choices and select the largest integer that lies within this intersection. This approach mirrors how inequality systems are handled in algebra and aptitude tests, where understanding overlapping conditions is important.

Step-by-Step Solution:
1) Solve the first inequality: 2x − 2(3 + 4x) < −1 − 2x. 2) Expand the bracket: 2x − 2 * 3 − 2 * 4x = 2x − 6 − 8x = −6 − 6x. 3) The inequality becomes −6 − 6x < −1 − 2x. 4) Add 6x to both sides: −6 < −1 + 4x. 5) Add 1 to both sides: −5 < 4x. 6) Divide by 4 (positive, so inequality direction stays): x > −5/4. 7) Now solve the second inequality: −1 − 2x > (−5/3) − (x/3). 8) Multiply every term by 3 to clear denominators: −3 − 6x > −5 − x. 9) Add 6x to both sides: −3 > −5 + 5x. 10) Add 5 to both sides: 2 > 5x. 11) Divide by 5: x < 2/5. 12) Combined, we have −5/4 < x < 2/5.

Verification / Alternative check:
The interval −5/4 < x < 2/5 corresponds approximately to −1.25 < x < 0.4. The integer values that lie in this open interval are x = −1 and x = 0. The question specifically asks for the greatest integer from the options that satisfies both inequalities. Among the options −2, −1, 0, 1, and 2, only −1 and 0 lie in the interval, and the greater of these two integers is 0. Therefore, x = 0 is the correct choice based on both inequalities and the wording of the question.

Why Other Options Are Wrong:
Option a (−2) is less than −1.25 and does not satisfy the first inequality. Option b (−1) is within the interval and is a valid solution, but it is not the greatest integer that works. Option d (1) and option e (2) are greater than 0.4 and violate the second inequality. Only option c (0) both satisfies the system and is the greatest integer among the choices that does so.

Common Pitfalls:
Students sometimes forget to compute the intersection of the two solution sets and instead choose any value that satisfies only one inequality. Another frequent mistake is mishandling signs when clearing denominators or moving terms across the inequality, or forgetting that multiplying by a negative number reverses the inequality direction. In this problem, we always divide or multiply by positive numbers, so the inequality directions remain unchanged. Carefully solving each inequality and then intersecting the results avoids these errors.

Final Answer:
The greatest integer among the given options that satisfies both inequalities is 0.