Question

b, c si d, daca se poate!!

Answer 1

b) - 8/(3x - 1) ∈ Z ⇔  (3x-1) | (-8)        3x - 1 ∈{-8,-4,-2,-1,1,2,4,8}
3x∈ {-7,-3,-1,0,2,3,5,9}    ⇒  x ∈{-1,0,1,3}
c)  (4x-2) /(5x +1)∈ Z  ⇔  (5x +1) | (4x - 2)
(5x+ 1) | (5x+1) ⇒  (5x+1) | 4(5x+1) = 20x +4  (1)
(5x+1) | (4x - 2) ⇒ (5x + 1) | 5(4x-2) = 20x - 10    (2)
⇒ (5x+1) | [(1) - (2)] = 14 ⇒  (5x+ 1) ∈ D14 
(5x + 1)∈ {-14,-7,- 2,-1 1,2,7,14}    5x ∈ {-15,-8,-3,-2,0,1,6,13}  ⇒ x ∈ {-3,0}
d)  (3x + 4) / (12x-1) ∈ N  ⇔   (12x - 1) | (3x + 4)
(12x - 1) | (12x - 1)     (1)
(12x - 1) | (3x + 4)⇒   (12x - 1) | 4(3x + 4) = 12x + 16     (2)
⇒ (12x -1) | [(2) - (1) ] = 17      (12x - 1) ∈ D17     12x - 1 ∈ {1,17}
12x ∈ {2,18}    ⇒ x∉ N