Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 2.7 Junior Find the answer

If for any $x \in \mathbb{R}$ , the inequality $|x| \geq ax$ always holds, then the range of the real number $a$ is

Pick one

Solution

When $x > 0$ , $x \geq ax$ always holds, which means $a \leq 1$ .

When $x = 0$ , $0 \geq a \times 0$ always holds, which means $a \in \mathbb{R}$ .

When $x < 0$ , $-x \geq ax$ always holds, which means $a \geq -1$ .

If for any $x \in \mathbb{R}$ , the inequality $|x| \geq ax$ always holds, then $-1 \leq a \leq 1$ .

Therefore, the correct choice is $\boxed{\text{B}}$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.