Question

|||x-1|-1|-1|=1 cate solutii reale are?

Answer 1

∨ (disjunctia) este echivalenta cu "SAU".

$\Big|\big||x-1|-1\big|-1\Big|=1 \\ \\ \big||x-1| - 1\big| - 1 = 1 ~~ \vee ~~ \big||x-1| - 1 \big|-1 = -1 \\ \\ \Leftrightarrow \\ \\ \big||x-1| - 1\big| = 2 ~~ \vee ~~ \big||x-1| - 1 \big| = 0 \\ \\ \Leftrightarrow \\ \\ \Big(|x-1|-1 = -2 ~~ \vee ~~ |x-1|-1 = 2\Big) ~~ \vee~~|x-1|-1 = 0 \\ \\ \Leftrightarrow (\text{disjunctia e asociativa)} \\ \\ |x-1|= -1 ~~ \vee ~~ |x-1| = 3 ~~ \vee ~~ |x-1| = 1 \\ \\ \Leftrightarrow (\text{modul din ceva nu poate fi negativ)}$

$x \in \emptyset ~~ \vee ~~ \big(x-1 = -3 ~~ \vee ~~ x-1 = 3\big) ~~ \vee~~ \\ \vee \big( x-1 = -1 ~~ \vee ~~ x-1 = 1) \\ \\ \Leftrightarrow\\ \\x \in \emptyset ~~ \vee ~~x = -2 ~~ \vee ~~x = 4 ~~\vee ~~x = 0~~ \vee ~~x = 2 \\ \\ \\ \Rightarrow S = \emptyset \cup \{-2\} \cup \{4\} \cup\{0\} \cup\{2\} \Rightarrow S = \Big\{-2,0,2,4\Big\} \\ \\ \Rightarrow \text{Ecuatia are 4 solutii}$

Answer 2

[tex]\it |||x-1|-1|-1|=1 \ \ \ \ (1) \\ \\ Notez\ |x-1| = a,\ (a>0) \ \ \ \ (2) \\ \\ (1),\ (2) \Rightarrow ||a-1|-1| =1 \ \ \ \ (3) \\ \\ Notez\ |a-1| = b \ \ \ \ (4)[/tex]


$\it (3),\ (4) \Rightarrow |b-1| =1 \Rightarrow b-1= \pm 1 \Rightarrow b-1 \in \{-1,\ 1\}|_{+1} \Rightarrow \\ \\ \Rightarrow b \in \{0,\ 2 \} \stackrel{(4)}{\Longrightarrow} |a-1| \in \{0,\ 2\} \Rightarrow a-1 \in \{-2,\ 0,\ 2\}|_{+1} \Rightarrow \\ \\ \stackrel{(2)}{\Longrightarrow} |x-1| \in \{1,\ 3 \} \Rightarrow x-1 \in\{ -3,\ -1,\ 1,\ 3\}|_{+1} \Rightarrow \\ \\ \Rightarfrow x \in\{ -2,\ 0,\ 2,\ 4\}$

Mulțimea soluțiilor ecuației date este :

S = {-2,  0,  2,  4}  și  card(S) = 4