MHB Solving the Floor Function Equation

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The equation |x| + √(⌊x⌋ + √(1 + {x})) = 1 is analyzed with the understanding that |x| ≤ 1, leading to x = 0 as one solution. It is established that {x} is non-negative, implying x cannot be positive. For the range -1 ≤ x < 0, the equation simplifies to |x| = -x, ⌊x⌋ = -1, and {x} = x + 1. This results in two solutions: one straightforward and another more complex. The discussion highlights the critical aspects of solving the floor function equation effectively.
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Solve the equation $\mid x \mid+\sqrt{\lfloor x \rfloor+\sqrt{1+\{x\}}} = 1$

where $\lfloor x \rfloor = $ floor function
 
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Obviously, |x| <= 1, and x = 0 is one solution. Further, {x} >= 0 for all x, so x cannot be positive. If -1 <= x < 0, then |x| = -x, floor(x) = -1 and {x} = x + 1. This gives an equation that has two solutions: one simple and one complicated.
 
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