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# Inequalities: Simultaneous Inequalities

If $$a^2 ≤ 16$$ and $$b^2 ≤ 36$$, which of the following inequalities gives all possible values of $$a+b$$?

Incorrect.

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Correct.

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Take both negative and positive values of $$a$$ and $$b$$ into account. $$a^2 ≤ 16 ~~~\rightarrow~~~ a ≥ -4$$ and $$a ≤ 4 ~~~\rightarrow~~~ -4 ≤ a ≤ 4$$ $$b^2 ≤ 36 ~~~\rightarrow~~~ b ≥ -6$$ and $$b ≤ 6 ~~~\rightarrow~~~ -6 ≤ b ≤ 6$$ Based on this, the minimum values of $$a$$ and $$b$$ are $$-4$$ and $$-6$$, and the maximum values are $$4$$ and $$6$$. Hence, the range of the sum of $$a$$ and $$b$$ is $$-10 ≤ a + b ≤ 10$$.

Incorrect.

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Incorrect.

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This answer choice is actually illogical, as $$-10$$ cannot be greater than or equal to $$10$$.

Incorrect.

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$$a+b \ge 10$$
$$a+b \le -10$$
$$-10 \le a+b \le 10$$
$$-10 \ge a+b \ge 10$$
$$a+b \ge 4$$