Question

[x] = 2 - x

[x + 1/2] = 2

[x + 1/2] = 2x + 7 totul supra 4

[...] = partea intreaga

Answer 1

Tinem cont ca
x≤[x]<x+1
atunci
x≤2-x<x+1
2x ≤2
x ≤1
1<2x
2x >1
x >1/2
deci
x∈ (1/2;1]dar [x]∈Z, 2∈Z deci x∈Z ..atunci (1/2;1]∩Z=1
x=1 solutie unica


2≤ x+1/2 <3

3/2 ≤x< 5/2

x∈[3/2;5/2)
aicinu avem restrictii deoarece 2∈Z




x+ 1/2 ≤ (2x +7)/4 <x +1/2+ 1

4x+2≤2x+7<4x+6

2x≤5
x≤5/2
si
1<2x
x>1/2

x∈(1/2;5/2]


dar cum[x+1/2]∈Z⇒(2x+7)/4∈Z
(2x/4+7/4) ∈Z
(x/2+1,75)∈Z
cautam doar valorile lui x din intervalul (1/2;5/2]
x/2+1,75=2
x/2=0,25..x=0,5∉(1/2;5/2]
x/2+1,75=3 ....x/2=1,25...x=2,5 ∈(1/2;5/2]
urmatoarele vor fi mai mari de 2,5 ,nu le mai incercam
x=2,5=5/2 solutie unica

verificare [2,5+0,5]=(2*2,5+7)/4=(5=7)/4=12/4=3 adevarat