Question

cine stie sa faca aceste probleme?

Answer 1

$2)~ |x-1| +|x-2| = 1 \\ \\ x-1 = 0 \Rightarrow x = 1, ~~ x - 2 =0 \Rightarrow x = 2\\ \\ \Rightarrow \text{Capetele noastre de intervale prin care o sa impartim expresia} \\ |x-1| + |x-2|\text{ vor fi 1 si 2}\\ \\ \Rightarrow |x-1|+|x-2|= \left\{ \begin{array}{ll} ?,~x\leq 1 \\ ?,~x\in [1,2]\\ ?,~x\geq 2 \end{array} \right \\ \\ \\ \Rightarrow |x-1|+|x-2|= \left\{ \begin{array}{ll} -(x-1)-(x-2) ,~x\leq 1 \\ (x-1)-(x-2),~x\in [1,2]\\ (x-1)+(x-2),~x\geq 2 \end{array} \right$$\boxed{1} ~~ x\leq 1: \\ \\ \Rightarrow -(x-1)-(x-2) = 1 \Rightarrow -2x +3 = 1 \Rightarrow -2x = -2 \\ \Rightarrow x = 1 \Rightarrow S_1 = \{1\} \cap (-\infty, 1] \Rightarrow S_1 = \{1\}\ \\ \\ \boxed{2}~~ x\in [1,2]: \\ \\ \Rightarrow (x-1)-(x-2) =1 \Rightarrow x-x-1+2 = 1 \Rightarrow 0\cdot x = 0 \Rightarrow \\\Rightarrow x \in \mathbb{R} \Rightarrow S_2 = \mathbb{R}\cap [1,2] \Rightarrow S_2= [1,2]$$\\ \\ \boxed{3}~~ x \geq 2: \\ \\ \Rightarrow (x-1)+(x-2) = 1 \Rightarrow 2x-3= 1 \Rightarrow 2x = 4 \Rightarrow x = 2 \Rightarrow \\ \Rightarrow S_3 = \{2\} \cap [2,+\infty) \Rightarrow S_3 = \{2\} \\ \\ \text{Din }\boxed{1} \cup \boxed{2} \cup \boxed{3}:\\ \\ \Rightarrow S = S_1 \cup S_2 \cup S_3 = \{1\} \cup [1,2] \cup \{2\} = [1,2] \\ \\ \Rightarrow \text{d) r\u{aspuns corect}}$