Solve the pair of linear inequalities 3x - 4 > 2 - x/3 and 3x + 5 > 4x - 5, and determine which of the following given values of x satisfies both inequalities simultaneously.

Introduction / Context:
Many aptitude exams include questions where you must solve two inequalities together and then verify which given option lies in the common solution set. This question involves linear inequalities with a fractional term and requires careful algebra. Once you solve each inequality separately, you combine their solution ranges and then test the candidate values of x.

Given Data / Assumptions:

• Inequality 1: 3x - 4 > 2 - x/3.
• Inequality 2: 3x + 5 > 4x - 5.
• We must find which among the listed values {7, 10, -11, 1, 0} satisfies both inequalities.

Concept / Approach:
The approach is to solve each inequality step by step using standard algebraic manipulations, taking care when multiplying or dividing by positive constants. Since there is no multiplication by negative numbers, the inequality signs will not flip. After obtaining each inequality's solution range, we intersect these ranges to find the common interval for x. Finally, we check which option lies inside that interval.

Step-by-Step Solution:
First inequality: 3x - 4 > 2 - x/3. Add x/3 to both sides: 3x + x/3 - 4 > 2. Write 3x as 9x/3: (9x/3 + x/3) - 4 > 2, so 10x/3 - 4 > 2. Add 4 to both sides: 10x/3 > 6. Multiply both sides by 3: 10x > 18. Divide by 10: x > 18/10 = 9/5 = 1.8. Second inequality: 3x + 5 > 4x - 5. Subtract 3x from both sides: 5 > x - 5. Add 5 to both sides: 10 > x, which is x < 10. Combine both: x > 1.8 and x < 10. So the common solution set is 1.8 < x < 10. Now test the options: 7 is between 1.8 and 10, so it satisfies both. 10 is not allowed because the second inequality is strict (>), and at x = 10 we have equality 3x + 5 = 4x - 5. -11, 1, and 0 are all less than or equal to 1.8, so they do not satisfy the first inequality.

Verification / Alternative check:
Substitute x = 7 into both inequalities. For the first: 3 * 7 - 4 = 21 - 4 = 17, and 2 - 7/3 = 2 - 2.333..., which is about -0.333..., so 17 > -0.333... holds. For the second: 3 * 7 + 5 = 21 + 5 = 26 and 4 * 7 - 5 = 28 - 5 = 23, so 26 > 23 holds. This confirms that x = 7 satisfies both inequalities.

Why Other Options Are Wrong:
At x = 10, the second inequality becomes 35 > 35, which is false because it is not strictly greater. Values -11, 1, and 0 fail the first inequality since they do not satisfy x > 1.8. Therefore none of those values belong to the common solution interval 1.8 < x < 10.

Common Pitfalls:
A common error is to forget to combine the solution sets and instead treat each inequality separately when checking options. Another frequent mistake is mishandling the fractional term x/3 or making arithmetic slips when simplifying. Always rewrite terms with a common denominator and carefully track each step when solving inequalities with fractions.

Final Answer:
The only option that satisfies both inequalities is 7.