Find all solutions in the integers of the equation ||x-3|-5| = 10, where the bars indicate absolute value.

Question

Mathematics

Roderick

12 January, 00:28

Find all solutions in the integers of the equation ||x-3|-5| = 10, where the bars indicate absolute value.

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Answers (1)

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Eddie Watts · Answer 1

x = 18 or x = - 12.

Step-by-step explanation:

||x-3|-5| = 10 only if |x-3|-5 = 10 or |x-3|-5 = - 10, i. e., if |x-3|=15 or |x-3|=-5; but |x-3| cannot be equal to - 5, because |x-3| should be a non-negative value. Therefore, the first equation is true only if |x-3|=15. |x-3| = 15 only if x-3 = 15 or x-3 = - 15, i. e., x = 18 or x = - 12. We can verify this in the following way: ||18-3|-5|=||15|-5|=|10|=10 and ||-12-3|-5|=||-15|-5|=|15-5|=|10|=10. This verify that our solution is correct.