Question

Determinati x apartine Z a.i : a) | x - 1| = 3 b) | x + 4 | = -1 c) | 1-x | = 4 d) | x- 1 |  <egal 3 e) | x + 2 | >egal 2  f) | x +5 | <egal2  g) | x -3 | > -3    barele sunt modul

Answer 1

a) x-1=3

x=4

x-1=-3

x=-2

b)x+4=-1

x=-5

x+4=1

x=-3

c) 1-x=4

-x=3

x=-3

1-x=-4

-x=-5

x=5

d) x-1≤3

x≤4

x∈(-∞,4]

x-1≤-3

x≤-2

x∈(-∞,-2]

e) x+2≥2

x≥0

x∈(+∞,0]

Answer 2

a) Deoarece | 3|=3 si | -3 | =3 avem 2 posibilitati
     x - 1 = 3 ⇒ x = 4
    x - 1 = - 3 ⇒ x = - 3+1= - 2
b) nu are solutie pentru ca modulul unui numar este  > 0 
c) 1 - x = 4 ⇒  - x = 4 - 1=3⇒x= - 3
   1 - x = - 4⇒ - x = - 4 - 1= - 5⇒x = 5
d) x - 1 <= 3⇒ x<= 4⇒x∈{0,1,2,3,4}
   -x + 1 <=3⇒ - x<=2⇒ - x∈{0,1,2}⇒x∈{0, - 1, - 2} 
e)x + 2>= 2⇒x>=0
  - x - 2>=2⇒ - x>= 4⇒ - x∈{4,5,6....,}⇒x∈{-4,-5,-6,....}
f) x+5<=2⇒x<= -3(2-5)⇒x∈{-3,-4,-5,-6,..}
  -x-5<=2⇒-x<=7⇒x∈{0,-1,-2,-3,-4,-5,-6,-7}
g)x-3≥ -3⇒x≥0
   -x+3≥-3⇒ - x ≥ - 6⇒x∈{6,5,4,3,2,1,0}