Question

VA ROG FRUMOSSSS
AM NEVOIE DE ACEST EX!!!!!PLS!!!!
E URGENT!!!
DAU 50 DE PUNCTE!!!
MERSIII​

Answer 1

Explicație pas cu pas:

a)

$|x| \leqslant 3 \iff - 3 \leqslant x \leqslant 3$

$I = \Big[-3;3\Big]$

$|x + 1| \geqslant 3$

$x + 1 \leqslant - 3 \iff x \leqslant - 4$

$x + 1 \geqslant 3 \iff x \geqslant 2$

$J = \Big(-\infty ; -4 \Big] \cup \Big[2 ; +\infty \Big)$

$\dfrac{3}{2} < \Big|x + \dfrac{1}{2} \Big|$

$x + \dfrac{1}{2} < - \dfrac{3}{2} \iff x < - \dfrac{3}{2} - \dfrac{1}{2} \\ x < - \dfrac{4}{2} \implies x < - 2$

$x + \dfrac{1}{2} > \dfrac{3}{2} \iff x > \dfrac{3}{2} - \dfrac{1}{2} \\ x > \dfrac{2}{2} \implies x > 1$

$K = \Big(-\infty ; -2 \Big) \cup \Big(1 ; +\infty \Big)$

b)

$I \cup J = \Big[-3;3\Big] \cup \Big(\Big(-\infty ; -4 \Big] \cup \Big[2 ; +\infty \Big)\Big) \\ = \Big(-\infty ; -4 \Big] \cup \Big[-3 ; +\infty \Big)$

$J \cup K = \Big(\Big(-\infty ; -4 \Big] \cup \Big[2 ; +\infty \Big)\Big) \cup \Big(\Big(-\infty ; -2 \Big) \cup \Big(1 ; +\infty \Big)\Big) \\ = \Big(-\infty ; -2 \Big) \cup \Big(1 ; +\infty \Big) = K$

$I \cap K = \Big[-3;3\Big] \cap \Big(\Big(-\infty ; -2 \Big) \cup \Big(1 ; +\infty \Big)\Big) \\ = \Big[-3; - 2\Big) \cup \Big(1;3\Big]$

$I \cap J = \Big[-3;3\Big] \cap \Big(\Big(-\infty ; -4 \Big] \cup \Big[2 ; +\infty \Big)\Big) \\ = \Big[2;3\Big]$

$I \cap J \cap K = \Big[2;3\Big]$

$I - J = \Big[-3;2\Big)$

$K - J = \Big(-4; -2\Big) \cup \Big(1; 2\Big)$

$\mathbb{R} - J = \Big(-4; 2\Big)$

complementara:

$\boxed {dac \breve a \ \ A \subseteq E \implies C_{E} A = E - A}$

$C_{\mathbb{R}} K = \mathbb{R} - K = \Big[-2;1\Big]$

$C_{\mathbb{R}} J = \mathbb{R} - J$