In aptitude inequalities, solve both inequalities together and choose a value of x that satisfies BOTH: 4(2x + 3) > 5 - x and 5x - 3(2x - 7) > 3x - 1.

Introduction / Context:
This question involves solving a system of two linear inequalities in one variable. You must find values of x that satisfy both inequalities simultaneously and then select the correct value from the options. Such problems test your ability to simplify inequalities, handle brackets, and understand how solution intervals overlap on the real number line.

Given Data / Assumptions:

• First inequality: 4(2x + 3) > 5 − x.
• Second inequality: 5x − 3(2x − 7) > 3x − 1.
• x is a real number.
• We must choose one value of x from the options that satisfies both inequalities.

Concept / Approach:
We solve each inequality separately by expanding brackets, collecting like terms, and isolating x. Each inequality yields an interval of solutions. The overall solution set is the intersection of these intervals, since both conditions must hold at the same time. Finally, we check the given discrete options and pick the value that lies within this intersection. This method is systematic and avoids guesswork.

Step-by-Step Solution:
1) Solve the first inequality: 4(2x + 3) > 5 − x. 2) Expand: 4 * (2x + 3) = 8x + 12. 3) The inequality becomes 8x + 12 > 5 − x. 4) Add x to both sides: 9x + 12 > 5. 5) Subtract 12 from both sides: 9x > −7, so x > −7/9. 6) Now solve the second inequality: 5x − 3(2x − 7) > 3x − 1. 7) Expand the bracket: 5x − 3(2x − 7) = 5x − 6x + 21 = −x + 21. 8) The inequality becomes −x + 21 > 3x − 1. 9) Add x to both sides: 21 > 4x − 1. 10) Add 1 to both sides: 22 > 4x, so x < 22/4 = 11/2. 11) Combine both results: x must satisfy x > −7/9 and x < 11/2, so the solution interval is (−7/9, 11/2).

Verification / Alternative check:
Now test each option. The interval (−7/9, 11/2) is approximately (−0.78, 5.5). Among the options −6, −1, 5, 6, and 'None of these', only x = 5 lies within this interval. Check x = 5 in the original inequalities. For the first: 4(2 * 5 + 3) = 4(10 + 3) = 4 * 13 = 52, and 5 − 5 = 0, so 52 > 0 is true. For the second: 5 * 5 − 3(2 * 5 − 7) = 25 − 3(10 − 7) = 25 − 9 = 16, and 3 * 5 − 1 = 15 − 1 = 14, so 16 > 14 is also true. Thus x = 5 satisfies both inequalities.

Why Other Options Are Wrong:
x = −6 and x = −1 are less than −7/9, so they fail the first inequality. For x = 6, which is greater than 11/2, the second inequality fails because it lies outside the allowed interval. Since one valid option exists, 'None of these' is not correct. Therefore, only x = 5 satisfies both inequalities simultaneously.

Common Pitfalls:
Students sometimes solve each inequality correctly but forget to take the intersection of the solution sets, or they mis-handle the expansion step and get incorrect coefficients. Another common mistake is to reverse the inequality sign when it is not required; in this problem we divide only by positive numbers, so the inequality direction remains the same. Drawing a quick number line with both bounds can help visualize the final interval clearly.

Final Answer:
The value of x that satisfies both inequalities is 5.