Difficulty: Medium
Correct Answer: -5
Explanation:
Introduction / Context:
This question tests compound inequalities connected by “and,” which means the final solution must satisfy both conditions simultaneously. The process is: solve each inequality separately to get two ranges for x, then take their intersection (overlap). Finally, because the question asks “which of the following values,” you compare each option against the intersection interval. These problems often trap students who accidentally take a union instead of an intersection or simplify one inequality incorrectly due to distribution mistakes.
Given Data / Assumptions:
Concept / Approach:
For each inequality:
1) Expand brackets correctly.
2) Collect x terms on one side and constants on the other.
3) Solve for x to get a range.
Then intersect the two ranges. After that, simply test the given options against the final range.
Step-by-Step Solution:
1) Solve Inequality 1:
2(3x - 2) < 6 - 3x
6x - 4 < 6 - 3x
6x + 3x < 6 + 4
9x < 10
x < 10/9
2) Solve Inequality 2:
6x + 2(6 - x) > 2x - 2
6x + 12 - 2x > 2x - 2
4x + 12 > 2x - 2
2x > -14
x > -7
3) Intersection of both results:
-7 < x < 10/9
4) Check options:
x = -5 lies between -7 and 10/9, so it satisfies both inequalities.
Verification / Alternative check:
Direct substitution for x = -5:
Inequality 1: 2(3(-5) - 2) = 2(-15 - 2) = -34, and 6 - 3(-5) = 21, so -34 < 21 (true).
Inequality 2: 6(-5) + 2(6 - (-5)) = -30 + 2(11) = -8, and 2(-5) - 2 = -12, so -8 > -12 (true).
Thus -5 works exactly.
Why Other Options Are Wrong:
• -8: fails x > -7.
• 5 and 8: fail x < 10/9.
• 2: also fails x < 10/9.
Common Pitfalls:
• Forgetting to distribute 2 across (3x - 2).
• Treating “and” as “or” and allowing values that satisfy only one inequality.
• Comparing options to only one inequality instead of the intersection.
Final Answer:
-5
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