Question

|3x+2| < |2-x|
Aflati solutia.

Answer 1

Daca 3x+2>=0 si 2-x>=0 (<=> x>=-2/3 si x<=2 <=> x apartine [-2/3,2]):
|3x+2|<|2-x| <=> 3x+2<2-x <=> 4x<0 <=> x<0. Avand in vedere si conditia x apartine [-2/3,2], deducem x apartine [-2/3,0).

Daca 3x+2>=0 si 2-x<0 (<=> x>=-2/3 si x>2 <=> x apartine (2,+inf)):
|3x+2|<|2-x| <=> 3x+2<x-2 <=> 2x<-4 <=> x<-2, ceea ce este imposibil deoarece x apartine (2,+inf).

Daca 3x+2<0 si 2-x>=0 (<=> x<-2/3 si x<=2 <=> x apartine (-inf,-2/3)):
|3x+2|<|2-x| <=> -(3x+2)<2-x <=> -3x-2<2-x <=> -4<2x <=> x>-2. Avand in vedere si conditia x apartine (-inf,-2/3), deducem ca x apartine (-2,-2/3).

Iar cazul 3x+2<0 si 2-x<0 este imposibil, deoarece asta ar insemna ca x<-2/3 si x>2, ceea ce-i absurd.

In concluzie, multimea solutiilor inecuatiei din enunt este [-2/3,0)U(-2,-2/3)=(-2,0).