Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 2.5 Junior Find the answer

If the solution set of the inequality $|x-4|-|x+5| \geq b$ about $x$ is empty, then the range of values for the real number $b$ is.

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

The geometric meaning of $|x-4|-|x+5|$ is the difference in distances on the number line from the point to 4 and to -5. The maximum difference is 9. If the solution set of the inequality $|x-4|-|x+5| \geq b$ about $x$ is empty, then the range of values for the real number $b$ is $b > 9$ .

Therefore, the answer is: $\boxed{b > 9}$ .

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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.