Задача

[Гост]

У?

[ammornil]

[tex]\begin{array}{|l} \frac{\normalsize{40}}{\normalsize{y+30}} \ge \frac{\normalsize{2}}{\normalsize{3}} \\ \\ \frac{\normalsize{40}}{\normalsize{y+30}} \le 1 \end{array}[/tex]

[tex]\text{ДМ }: y+30 \ne 0 \Leftrightarrow y\ne -30 \Rightarrow y \in (-\infty; -30) \cup (-30; +\infty)[/tex]

(1 случай)
[tex]\underline{y+30<0 \Leftrightarrow y<-30}[/tex]

[tex]\begin{array}{|l} \frac{\normalsize{40}}{\normalsize{y+30}} \ge \frac{\normalsize{2}}{\normalsize{3}} \hspace{0.7em} |\cdot{\frac{3}{2}(y+30)<0}\\ \\ \frac{\normalsize{40}}{\normalsize{y+30}} \le 1 \hspace{0.7em} |\cdot{(y+30)<0} \end{array} \Leftrightarrow \begin{array}{|l} 40\cdot{\frac{\normalsize{3}}{\normalsize{2}}} \le y+30 \\ 40 \ge y+30 \end{array} \Leftrightarrow \begin{array}{|l} y \ge 30 \\ y \le 10 \end{array} \Rightarrow y \in \varnothing[/tex]

(2 случай)
[tex]\underline{y+30>0 \Leftrightarrow y>-30}[/tex]

[tex]\begin{array}{|l} \frac{\normalsize{40}}{\normalsize{y+30}} \ge \frac{\normalsize{2}}{\normalsize{3}} \hspace{0.7em} |\cdot{\frac{3}{2}(y+30)>0}\\ \\ \frac{\normalsize{40}}{\normalsize{y+30}} \le 1 \hspace{0.7em} |\cdot{(y+30)>0} \end{array} \Leftrightarrow \begin{array}{|l} 40\cdot{\frac{\normalsize{3}}{\normalsize{2}}} \ge y+30 \\ 40 \le y+30 \end{array} \Leftrightarrow \begin{array}{|l} y \le 30 \\ y \ge 10 \end{array} \Rightarrow y \in [10; 30][/tex] $$ y \in [10; 30] $$

Друг начин без отделни случаи:

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[tex]\begin{array}{|l} \frac{\normalsize{40}}{\normalsize{y+30}} \ge \frac{\normalsize{2}}{\normalsize{3}} \hspace{0.7em} |\cdot{\frac{3}{2}(y+30)^{2}>0}\\ \\ \frac{\normalsize{40}}{\normalsize{y+30}} \le 1 \hspace{0.7em} |\cdot{(y+30)^{2}>0} \end{array} \Leftrightarrow \begin{array}{|l} 40\cdot{\frac{\normalsize{3}}{\normalsize{2}}}\cdot{(y+30)} \ge (y+30)^{2} \\ 40(y+30) \le (y+30)^{2} \end{array} \Leftrightarrow \begin{array}{|l} 60y+1800 \ge y^{2} +60y +900 \\ 40y+1200 \le y^{2}+60y+900 \end{array} \Leftrightarrow \begin{array}{|l} y^{2}-900 \le 0 \\ y^{2}+20y-300 \ge 0 \end{array} \Leftrightarrow[/tex]

[tex]\Leftrightarrow \begin{array}{|l} (y-30)(y+30) \le 0 \\ (y-10)(y+30) \ge 0 \end{array}[/tex]$$ y \in [10; 30] $$
[tex][/tex]

Screenshot 2023-05-14 180934.png (3.44 KiB) Прегледано 362 пъти