Question

rezolvati inecuatia:
|x+2|×(|x+3|-2)<0
va rog repede!

Answer 1

|x+2|≥0
cum
|x+2|×(|x+3|-2)<0
trebui eliminat valoarea pt care |x+2|=0 si anume x=-2
deci indiferent care va fi solutia inecuatiei, din ea trebuie sa scadem (in sensul scaderii ca  operatie cu multimi) multimea {-2}
ramane acum sa rezolvam

|x+3|-2<0
adica
|x+3|<2
-2<x+3<2 |-3
-5<x<-1
x∈(-5;-1) dar sa nu uitam sa eliminam pe {-2}
deci
x∈(-5;-1)\{-2}  sau, altfel scris
x∈(-5;-2)∪(-2;-1)

as tricky as that!!!

Answer 2

[tex]\it |x + 2| (|x + 3| - 2) \ \textless \ 0 \ \ \ \ (1) \\ \\ |x + 2| \geq0 \ \ \ \ \ (2) \\ \\ (1), (2) \Rightarrow \begin{cases} \it x+2 \ne 0 \Rightarrow x \ne -2 \ \ \ \ (3) \\ \\ \it |x+3|-2 \ \textless \ 0 \Rightarrow |x+3| \ \textless \ 2 \ \ \ \ \ (4) \end{cases}[/tex]

$\it (4) \Rightarrow -2\ \textless \ x+3\ \textless \ 2|_{-3} \Rightarrow -5\ \textless \ x\ \textless \ -1 \Rightarrow x \in (-5,\ -1) \ \ \ \ (5) \\ \\ (3), (5) \Rightarrow x \in (-5,\ -2) \cup (-2,\ -1)$