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Olympiad workbook
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Olympiad workbook
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IOQM
Chapter
Basics of Mathematics

Olympiad workbook · IOQM · Basics of Mathematics

Find the number of integer solutions to ||x|-2020| < 5 .

Find the number of integer solutions to ∣∣x∣−2020∣<5||x|-2020| < 5.

Correct answer

18

Step-by-step solution

∣∣x∣−2020∣<5⇒−5<∣x∣−2020<5⇒2015<∣x∣<2025⇒−2025<x<−2015 OR 2015<x<2025⇒x=−2024,−2023,…..,−2016; OR x=2016,2017,….,2024 Number of integer solutions =18\begin{aligned} & ||x|-2020| < 5 \\ & \Rightarrow \quad-5 < |x|-2020 < 5 \\ & \Rightarrow \quad 2015 < |x| < 2025 \\ & \Rightarrow \quad-2025 < x < -2015 \text { OR } 2015 < x < 2025 \\ & \Rightarrow \quad x=-2024,-2023, \ldots . .,-2016 ; \\ & \text { OR } x=2016,2017, \ldots ., 2024 \\ & \text { Number of integer solutions }=18\end{aligned}
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