Explicație pas cu pas:
g)
[tex]\frac{2x - 3}{x + 5} = \frac{2x + 10 - 13}{x + 5} = \frac{2(x + 5) - 13}{x + 5} = 2 - \frac{13}{x + 5}\in \mathbb{Z} \\ \implies \frac{13}{x + 5}\in \mathbb{Z} \iff (x + 5) ∈ \{ - 13; - 1;1;13 \} \\ [/tex]
[tex]x + 5 = - 13 \implies x = - 18 \\ x + 5 = - 1 \implies x = - 6 \\ x + 5 = 1 \implies x = - 4 \\ x + 5 = 13 \implies x = 8[/tex]
[tex]\implies x ∈ \{ - 18; - 6; - 4;8 \}[/tex]
h.
[tex]{x}^{2} + x = 2 \cdot (1 + 2 + 3 + ... + 2021) \\ x(x + 1) = 2 \cdot \frac{2021 \cdot 2022}{2} \\ x(x + 1) = 2021 \cdot 2022 \\ x(x + 1) = 2021 \cdot (2021 + 1) \\x \in \mathbb{N} \implies x = 2021 \implies \bf H = \{2021\}[/tex]
