In aptitude inequalities, solve the chain inequality correctly and choose a value of x that satisfies BOTH conditions: 5x - 3(2x - 7) > 3x - 1 and 3x - 1 < 7 + 4x.

Introduction / Context:
This question involves a pair of linear inequalities in one variable that must both be satisfied. Such chain inequalities are common in aptitude tests and check your ability to simplify inequalities, distribute correctly, and combine solution sets using interval intersection. You must then pick a specific value of x from the options that lies within the common solution region.

Given Data / Assumptions:

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

Concept / Approach:
We simplify each inequality step by step to isolate x. The first inequality will give an upper bound or lower bound on x, and the second inequality will give another bound. The intersection of these bounds determines the full solution set. Finally, we test each option to see which values lie in this intersection and satisfy both inequalities together.

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

Verification / Alternative check:
Now evaluate each option against the interval (−8, 11/2), which is approximately (−8, 5.5). Among the options 6, 9, −6, −9, and 'None of these', only x = −6 lies within the interval. Test x = −6 in both inequalities. First: 5(−6) − 3(2(−6) − 7) = −30 − 3(−12 − 7) = −30 − 3(−19) = −30 + 57 = 27, and the right side is 3(−6) − 1 = −18 − 1 = −19, so 27 > −19, which is true. Second: 3(−6) − 1 = −18 − 1 = −19, and 7 + 4(−6) = 7 − 24 = −17, so −19 < −17, which is also true. Thus x = −6 satisfies both inequalities.

Why Other Options Are Wrong:
Options 6 and 9 are greater than 11/2 (5.5) and therefore violate the first inequality, which requires x < 11/2. Option −9 is less than −8 and violates the second inequality, which requires x > −8. Since we have found that x = −6 is a valid solution, the option 'None of these' is incorrect. Only x = −6 lies in the intersection and satisfies both inequalities.

Common Pitfalls:
Common mistakes include sign errors when expanding −3(2x − 7), forgetting to change inequality signs when appropriate (though in this problem we never divide by a negative number), and failing to correctly interpret the combined inequality as an intersection of intervals. Drawing a quick number line and marking both bounds helps visualise the final solution and can prevent misreading of the interval endpoints.

Final Answer:
The value of x that satisfies both inequalities is -6.