Question

Ajutor..Am nevoie urgent..In maxim 2 ore..​

Answer 1

a)

$I = \left\{x\in \mathbb{R}\big|\,\, |x|\leq 3\right\} \\\\-3\leq x\leq 3 \Rightarrow I = [-3,3]\\ \\\\J = \left\{x\in \mathbb{R}\big|\,\,|x+1|\geq 3\right\} \\ \\ -3\geq x+1\quad \vee\quad x+1\geq 3 \\ -4\geq x\quad \vee\quad x \geq 2\\ x\leq -4\quad \vee\quad x\geq 2 \\ \\ \Rightarrow J = [-\infty, -4]\cup[2,+\infty)$

$K = \left\{x\in \mathbb{R}\Bigg|\,\,\dfrac{3}{2}<\left|x+\dfrac{1}{2}\right|\right\} \\ \\\\ -\dfrac{3}{2}>x+\dfrac{1}{2}\quad \vee\quad x+\dfrac{1}{2} >\dfrac{3}{2} \\ \\ -2 >x\quad \vee \quad x > 1\\ x<-2\quad \vee\quad x>1 \\ \\ \Rightarrow K = (-\infty, -2)\cup(1,+\infty)$

b)

$I = [-3,3]\\J = (-\infty, -4)\cup[2,+\infty)\\K = (-\infty, -2)\cup(1,+\infty)\\ \\ \ I\cup J = \mathbb{R}\,\backslash\,[-4,-3)\\ J\cup K = \mathbb{R}\,\backslash\, [-2,1]\\ I\cap K =[-3,2)\cup (1, 3]\\ I \cap J = [2, 3]\\ I\cap J\cap K = [2, 3]$

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