Inequalities: Inequalities Involving an Absolute Value - the Variable Case
If $$|x| ≥ x+2$$, which of the following must be true?
Incorrect.
[[snippet]]
Try plugging in $$x = -5$$:
>$$|{-5}| ≥ -5 + 2$$
>$$5 ≥ -3$$
While $$-5$$ works for the inequality in the question stem, it is not between $$-1$$ and $$1$$. Therefore, this answer choice is _not necessarily true_, and it should be eliminated.
Incorrect.
[[snippet]]
Try plugging in $$x = -5$$:
>$$|{-5}| ≥ -5 + 2$$
>$$5 ≥ -3$$
Although $$-5$$ works for the inequality in the question stem, since $$-5 \le -4$$,
you cannot yet eliminate this answer choice.
Try plugging in $$x = -2$$:
>$$|{-2}| ≥ -2 + 2$$
>$$2 ≥ 0$$
While $$-2$$ works for the inequality in the question stem, it is not $$\le -4$$. Therefore, this answer choice is _not necessarily true_, and it should be eliminated.
Correct.
[[snippet]]
Here's a possible elimination scenario: Try plugging in $$x = -5$$:
>$$|{-5}| ≥ -5 + 2$$
>$$5 ≥ -3$$
Although $$-5$$ works for the inequality in the question stem, since $$-5 \le -1$$, you cannot yet eliminate this answer choice. However, $$x = -5$$ does eliminate answer choices A, B, and E.
Try plugging in $$x = -2$$:
>$$|{-2}| ≥ -2 + 2$$
>$$2 ≥ 0$$
Although $$-2$$ works for the inequality in the question stem, since $$-2 \le -1$$, you still cannot eliminate this answer choice. However, $$x = -2$$ eliminates answer choice D.
By process of elimination, this answer choice (C) is the only one left standing. Therefore, it is correct.
Incorrect.
[[snippet]]
Try plugging in $$x = -5$$:
>$$|{-5}| ≥ -5 + 2$$
>$$5 ≥ -3$$
While $$-5$$ works for the inequality in the question stem, it is not $$\ge 2$$. Therefore, this answer choice is _not necessarily true_, and it should be eliminated.
Incorrect.
[[snippet]]
Try plugging in $$x = -5$$:
>$$|{-5}| ≥ -5 + 2$$
>$$5 ≥ -3$$
While $$-5$$ works for the inequality in the question stem, it is not greater than 1. Therefore, this answer choice is _not necessarily true_, and it should be eliminated.
$$x \ge 1$$
$$x \ge 2$$
$$x \le -1$$
$$x \le -4$$
$$-1 < x < 1$$
