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.
Analytic solution then computer verification.
Let x = i + f
where i is ⌊x⌋ and f is {x}
Case 1: x = 0; 0 is a solution
Case 2: x > 0
⌊x⌋ + 2{-x} = 3x becomes
i + 2*(1-f) = 3i + 3f
2 = 2i + 5f --> i=0, f=2/5
x = 0.4
Case 3: x < 0
x=-(i+f), ⌊x⌋=-(i+1), {-x}=f
-(i+1) + 2f = -3(i+f)
-1 = -2i - 5f
if i=0, f=0.2
x = -(i+f) = -0.2
Solutions: -0.2, 0, 0.4
------------
import math
epsilon = .00000001
for i in range(-100000,100000):
n = i/1000
if abs(math.floor(n) + 2*((-n)%1) - 3*n) < epsilon:
print(n)
Output:
-0.2
0.0
0.4
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Posted by Larry
on 2023-09-14 11:27:25 |
solution.
