In aptitude simplification with linear inequalities, solve the compound inequality and choose the value of x that satisfies both conditions: 3(2 - 3x) < (2 - 3x) and (2 - 3x) ≥ (4x - 6).

Introduction / Context:
This question belongs to aptitude simplification with linear inequalities in one variable. You are given a pair of inequalities that must be satisfied at the same time, and you must choose a value of x from the options that satisfies both. Problems like this test your ability to manipulate inequalities, handle brackets, and understand how solution intervals overlap.

Given Data / Assumptions:

• First inequality: 3(2 - 3x) < (2 - 3x).
• Second inequality: (2 - 3x) ≥ (4x - 6).
• x is a real number.
• We must choose one option that satisfies both inequalities simultaneously.

Concept / Approach:
For each inequality, we remove brackets, simplify, and isolate x. The solution to each inequality is an interval on the real line. The combined solution must satisfy both, so we take the intersection of these intervals. Finally, we test the given discrete options against this intersection and identify which value of x fits the combined condition. This systematic approach avoids guesswork and ensures an accurate result.

Step-by-Step Solution:
1) Start with 3(2 - 3x) < (2 - 3x). 2) Expand the left side: 3 * (2 - 3x) = 6 - 9x. 3) The inequality becomes 6 - 9x < 2 - 3x. 4) Subtract 2 from both sides: 4 - 9x < -3x. 5) Add 9x to both sides: 4 < 6x, so x > 4/6 = 2/3. 6) Now consider (2 - 3x) ≥ (4x - 6). 7) Add 3x to both sides: 2 ≥ 4x + 3x - 6 = 7x - 6. 8) Add 6 to both sides: 8 ≥ 7x, so x ≤ 8/7. 9) Combine the results: x must satisfy x > 2/3 and x ≤ 8/7. So the solution interval is (2/3, 8/7]. 10) Among the options −2, −1, 1, 2, and 'None of these', only x = 1 lies in the interval (2/3, 8/7].

Verification / Alternative check:
Substitute x = 1 into the original inequalities. For the first inequality, 3(2 - 3 * 1) = 3(2 - 3) = 3 * (−1) = −3 and (2 - 3 * 1) = −1, so −3 < −1, which is true. For the second inequality, (2 - 3 * 1) = −1 and (4 * 1 - 6) = 4 - 6 = −2, so −1 ≥ −2, which is also true. Thus x = 1 satisfies both inequalities exactly, confirming that it is a valid solution and consistent with the intersection found earlier.

Why Other Options Are Wrong:
For x = −2 or x = −1, the first inequality fails because 3(2 - 3x) becomes larger than (2 - 3x) when evaluated numerically. For x = 2, the second inequality fails because (2 - 3 * 2) is less than (4 * 2 - 6). Therefore none of these values satisfy both inequalities. The option 'None of these' is wrong because we have already found that x = 1 is a valid solution.

Common Pitfalls:
A common mistake is to divide or multiply both sides of an inequality by an expression involving x without checking its sign, which can reverse the inequality direction incorrectly. Another error is failing to compute the intersection of the solution sets and choosing a value that only satisfies one inequality. Carefully isolating x in each inequality and then intersecting the resulting intervals avoids these issues.

Final Answer:
The value of x that satisfies both inequalities is 1.