Question

Vă rog frumos să mă ajutați!

Answer 1

[tex]\it \Big|\dfrac{2n+1}{3}\Big|\ \textless \ 2 \Rightarrow -2\ \textless \ \dfrac{2n+1}{3}\ \textless \ 2|_{\cdot3} \Rightarrow -6\ \textless \ 2n+1\ \textless \ 6|_{-1} \Rightarrow \\\;\; \\\;\\ -7\ \textless \ 2n\ \textless \ 5|_{:2} \Rightarrow -\dfrac{7}{2}\ \textless \ n\ \textless \ \dfrac{5}{2} \Rightarrow -3\dfrac{1}{2}\ \textless \ n\ \textless \ 2\dfrac{1}{2} \\\;\\ \\\;\\ Pentru\ \ n\ \in\mathbb{Z} \Longrightarrow n\ \in \{-3,\ -2,\ -1,\ 0,\ 1,\ 2\}[/tex]

2) [-4,16] = - 5 (am luat întregul cel mai apropiat, dar mai mic decât  - 4,16)

{-4,16} = -4,16 - [4,16] =-4,16 - (-5) = -4,16 + 5 = 5 - 4,16 =0, 84

3)

[tex]\it \dfrac{1}{2} \ \textless \ \dfrac{a+1}{2}-\dfrac{3}{2}|_{\cdot2} \Rightarrow 1\ \textless \ a+1-3 \Rightarrow 1-1+3\ \textless \ a \Rightarrow 3\ \textless \ a\Rightarrow a\ \textgreater \ 3 \Rightarrow \\\;\\ \\\;\\ \Rightarrow a\in\{4, \ 5,\ 6,\ 7,\ ...\}[/tex]

4)

[tex]\it \dfrac{10}{2x+5} \in\mathbb{Z} \Rightarrow 2x+5 | 10 \Rightarrow 2x+5 \in D_{10} \Rightarrow \\\;\\ \\\;\\ \Rightarrow2x+5\in\{\pm1,\ \pm2,\ \pm5,\ \pm10\}\Rightarrow \\\;\\ \\\;\\ \Rightarrow 2x+5\in\{-10,\ -5,\ -2,\ -1,\ 1,\ 2,\ 5,\ 10\}|_{-5} \Rightarrow[/tex]

$\it \Rightarrow 2x\in\{-15,\ -10,\ -7,\ -6,\ -4,\ -3,\ 0,\ 5\}$

$\it Pentru \ x\in\mathbb{Z}\ \Rightarrow x\in\{-5,\ -3,\ -2,\ 0\}$

La final, am ales din mulțimea precedentă numai numerele care se împart exact

la 2.

5)

[tex]\it \dfrac{6}{2x-3} \in\mathbb{Z} \Rightarrow 2x-3 | 6 \Rightarrow 2x-3 \in D_{6} \Rightarrow \\\;\\ \\\;\\ \Rightarrow2x-3\in\{\pm1,\ \pm2,\ \pm3,\ \pm6\}\Rightarrow \\\;\\ \\\;\\ \Rightarrow 2x-3\in\{-6,\ -3,\ -2,\ -1,\ 1,\ 2,\ 3,\ 6\}|_{+3} \Rightarrow [/tex]

$\it \Rightarrow 2x\in\{-3,\ 0,\ 1,\ 2,\ 4,\ 5,\ 6,\ 9\}$

$\it Pentru \ x\in\mathbb{Z}\ \Rightarrow x\in\{0,\ 1,\ 2,\ 3\}$

La final, am ales din mulțimea precedentă numai numerele care se împart exact

la 2.