If k ≤ x ≤ 3k + 12, which of the following must be true? I. x - 12 ≤ 3k II. k ≥ -6 III. x - k ≥ 0
- I only
- I and II only
- II and III only
- I, II, and III
Explanation
Let's analyze the given inequality: k ≤ x ≤ 3k + 12
We can derive the following conclusions:
I. x - 12 ≤ 3k: Subtracting 12 from all parts of the inequality gives x - 12 ≤ 3k - 12 + 12, which simplifies to x - 12 ≤ 3k. (True)
II. k ≥ -6: Since k ≤ x and x ≤ 3k + 12, we can substitute x for k in the second part of the inequality, getting k ≤ 3k + 12. Subtracting 3k from both sides gives -2k ≤ 12, and dividing by -2 (which flips the inequality sign) gives k ≥ -6. (True)
III. x - k ≥ 0: Since k ≤ x, we can subtract k from both sides to get x - k ≥ 0. (True)
Since all three statements are true, the correct answer is: I, II, and III
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