If 7x + 2 ≥ x - 2 and 7 + 2x ≥ 3 + 3x, then which of the following values of x satisfies both inequalities?

Introduction / Context:
This question involves solving two linear inequalities in x and then identifying which option satisfies both simultaneously. The idea is to simplify each inequality to obtain a range of possible x values. The actual solution set is the intersection of these ranges, and the correct option must lie within this final interval.

Given Data / Assumptions:

• Inequality 1: 7x + 2 ≥ x - 2.
• Inequality 2: 7 + 2x ≥ 3 + 3x.
• We need to select an x from the options that satisfies both inequalities.

Concept / Approach:
The general approach is to solve each inequality by standard algebraic manipulation. For each one, collect all x terms on one side and constants on the other. This converts each inequality into a simple inequality such as x ≥ some value or x ≤ some value. Then intersect the two solution sets to get the overall allowed range for x. Finally, check which given option lies in this interval.

Step-by-Step Solution:

Verification / Alternative check:
Substitute x = 3 into both inequalities. For the first: 7*3 + 2 = 21 + 2 = 23 and x - 2 = 3 - 2 = 1, so 23 ≥ 1 is true. For the second: 7 + 2*3 = 7 + 6 = 13 and 3 + 3*3 = 3 + 9 = 12, so 13 ≥ 12 is also true. Thus, x = 3 satisfies both conditions simultaneously.

Why Other Options Are Wrong:

Common Pitfalls:
Errors may occur when handling negative numbers, such as forgetting to divide correctly or miscalculating -4/6. Another mistake is not considering both inequalities together and choosing a value that satisfies only one of them. Always intersect the solution sets and then verify by direct substitution into the original inequalities.

Final Answer:
3