solve-2x-1-3-x-2-x-

Question Number 87536 by M±th+et£s last updated on 04/Apr/20

s o l v e

∣ 2 x - 1 ∣= 3 ⌊ x ⌋ + 2 {x}

Answered by mr W last updated on 05/Apr/20

x = n + δ

⌊ x ⌋ = n

{x} = δ

2 x - 1 = 2 n + 2 δ - 1

i f 2 x - 1 ⩾ 0, i . e . x ⩾ \frac{1}{2} :

2 n + 2 δ - 1 = 3 n + 2 δ

\Rightarrow 2 n - 1 = 3 n \Rightarrow n = - 1

\Rightarrow δ = a n y v a l u e 0 ⩽ δ < 0

\Rightarrow - 1 ⩽ x < 0 \Rightarrow n o s o l u t i o n

i f 2 x - 1 < 0, i . e . x < \frac{1}{2} :

- 2 n - 2 δ + 1 = 3 n + 2 δ

5 n + 4 δ = 1

4 δ m u s t b e i n t g e r .

δ = 0 : 5 n = 1 \Rightarrow n o s o l u t i o n

δ = \frac{1}{4} : 5 n = 0 \Rightarrow n = 0 \Rightarrow x = \frac{1}{4} < \frac{1}{2} \Rightarrow s o l u t i o n

δ = \frac{1}{2} : 5 n = - 1 \Rightarrow n o s o l u t i o n

δ = \frac{3}{4} : 5 n = - 2 \Rightarrow n o s o l u t i o n

\Rightarrow o n l y s o l u t i o n i s x = \frac{1}{4} .

Commented by M±th+et£s last updated on 05/Apr/20

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