Determine the total number of real values of x satisfying this equation:
⌊x⌋ + 2 {-x} = 3x
Notes:
• ⌊N⌋ is floor of N, which is equal to the greatest integer less than or equal to N.
{N} = N - ⌊N⌋.
• Computer program/excel solver assisted solutions are welcome, but a semi-analytic (hand calculator and p&p) methodology is preferred.
-floor(-x)=ceil(x) which for nonintegers = floor(x)+1
and for integers = x
{-x} = -x - floor(-x) = -x+floor(x)+1 for nonintegers and for integers =0
so for integers the original equation can be written
floor(x)=3x
which has integers solution 0 (and a noninteger -1/3 that we reject)
for nonintegers the original equation can be written
floor(x) + 2(-x+floor(x)+1)=3x
3floor(x)+2=5x
which has noninteger solutions -1/5 and 2/5 (and an integer solution 1 that we reject.)
So there are 3 solutions x={-1/5, 0, 2/5}
https://www.desmos.com/calculator/kkubum2ocj
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Posted by Jer
on 2023-09-14 10:39:39 |
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