Задача 62880 ...

6182340a3d5d7c0560db11d7

27.01.2022 10:59:23

решить неравенство :_(log)0.5 (x-4/(x+3)) ≥ -2 показать на графике .

5f3ea7e3faf909182968ddd9

27.01.2022 13:03:20

★

[m]log_{0,5}\frac{x-4}{x+3} ≥ -2[/m]

[m]log_{0,5}\frac{x-4}{x+3} ≥ -2\cdot log_{0,5} 0,5[/m]

[m]log_{0,5}\frac{x-4}{x+3} ≥ log_{0,5} 0,5^{-2}[/m]

[m]log_{0,5}\frac{x-4}{x+3} ≥ log_{0,5} 4[/m]


Логарифмическая функция убывающая, поэтому

[m]0 <\frac{x-4}{x+3} ≤ 4[/m]

[m]\left\{\begin {matrix}\frac{x-4}{x+3}>0\\\frac{x-4}{x+3} ≤ 4\end {matrix}\right.[/m] ⇒ [m]\left\{\begin {matrix}\frac{x-4}{x+3}>0\\\frac{x-4}{x+3} -4≤ 0\end {matrix}\right.[/m] ⇒ [m]\left\{\begin {matrix}\frac{x-4}{x+3}>0\\\frac{x-4-4(x+3)}{x+3} ≤ 0\end {matrix}\right.[/m]

[m]\left\{\begin {matrix}\frac{x-4}{x+3}>0\\\frac{-3x-16}{x+3} ≤ 0\end {matrix}\right.[/m]

[m]\left\{\begin {matrix}\frac{x-4}{x+3}>0\\\frac{3x+16}{x+3} ≥ 0\end {matrix}\right.[/m]

[m]\left\{\begin {matrix}x+3>0\\x-4 >0\\3x+16 ≥ 0\end {matrix}\right.[/m] или [m]\left\{\begin {matrix}x+3<0\\x-4 <0\\3x+16 ≤ 0\end {matrix}\right.[/m]

[m]x>4 [/m] или [m]x ≤ -\frac{16}{3}[/m]

О т в е т. [m](- ∞ ; -\frac{16}{3}]\cup(4;+ ∞ )[/m]