In each of the following exercises, verify the given statement carefully, proceeding step by step. Validate each step that involves an inequality by using some statement found in this section.
Let x ∈ R.
a) Prove that |x| ≤ 2 implies |x2 − 4| ≤ 4|x − 2|.
b) Prove that |x| ≤ 1 implies |x2 + 2x − 3| ≤ 4|x − 1|.
c) Prove that −3 ≤ x ≤ 2 implies |x2 + x − 6| ≤ 6|x − 2|.
d) Prove that −1 < x < 0 implies |x3 − 2x + 1| < 1.25|x − 1|.
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