Find the number of integer solutions to ||x| - 2020|

Question

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

Accepted Answer

{"type":"doc","content":[{"content":[{"text":"The inequality is given by:","type":"text"}],"type":"paragraph"},{"attrs":{"latex":"||x| - 2020| < 5"},"type":"math_block"},{"content":[{"text":"This is equivalent to:","type":"text"}],"type":"paragraph"},{"attrs":{"latex":"-5 < |x| - 2020 < 5 \implies 2015 < |x| < 2025"},"type":"math_block"},{"type":"paragraph","content":[{"type":"text","text":"Since "},{"attrs":{"latex":"x"},"type":"math_inline"},{"text":" must be an integer, ","type":"text"},{"attrs":{"latex":"|x|"},"type":"math_inline"},{"text":" must also be an integer. The possible values of ","type":"text"},{"attrs":{"latex":"|x|"},"type":"math_inline"},{"text":" are:","type":"text"}]},{"attrs":{"latex":"|x| \in \{2016, 2017, 2018, 2019, 2020, 2021, 2022, 2023, 2024\}"},"type":"math_block"},{"content":[{"type":"text","text":"There are exactly 9 possible positive integer values for "},{"attrs":{"latex":"|x|"},"type":"math_inline"},{"text":".","type":"text"}],"type":"paragraph"},{"content":[{"type":"text","text":"For each value of "},{"attrs":{"latex":"|x|"},"type":"math_inline"},{"text":", there are 2 solutions for ","type":"text"},{"attrs":{"latex":"x"},"type":"math_inline"},{"text":" (one positive and one negative).","type":"text"},{"type":"hard_break"},{"text":"Thus, the total number of integer solutions is:","type":"text"}],"type":"paragraph"},{"type":"math_block","attrs":{"latex":"\text{Total} = 9 \times 2 = 18"}}]}

Guide / Hint

Solution