Question

Rezolvati in Z ecuatiile:
a) modul de (x-2) = modul de (2x-3);
b) modul de (x-2) + modul de (y-3) =1.
Multumesc!

Answer 1

a)|(x-2)|=|(2x-3)|
2-x=3-2x
3+2=x+2x
5=3x=>x=5/3
b)|(x-2)|+|(y-3)|=1
2-x+3-y=1
2+3+1=xy
6=xy

Answer 2

Se expliciteaza modulele (un tabel este foarte benefic).

Distingem cazurile:

I) x∈(-∞, 3/2) ⇒ ecuatia devine:

-x + 2 = -2x + 3 ⇒ x  = 1∈ (-∞, 3/2)

II) x∈ [3/2, 2 ] ⇒ ecuatia devine: 

-x+2 = 2x - 3 ⇒ x = 5/3 ∈ [ 3/2,  2]

III) x ∈ (2,  ∞) ⇒ ecuatia devine :

x - 2 = 2x -3 ⇒ x = 1 ∉ (2, ∞) ⇒ ecuatia nu are solutie in acest interval.

Deci, ecuatia admite doua solutii : x =  1,  x=  5/3