Question

Примеры на фото, помогите с решением любых.

Answer 1

$1)\; \; \frac{(5-2x)(x+3)}{(2x-7)(6-5x)}\geq 0\; \; ,\; \; \; \frac{(2x-5)(x+3)}{(2x-7)(5x-6)}\geq 0\\\\znali:\; \; +++[-3\, ]---(1,2)+++[\, 2,5\, ]---(3,5)+++\\\\\underline {x\in (-\infty ;-3\, ]\cup (1,2\, ;\, 2,5\, ]\cup (3,5\, ;\, +\infty )}$

$2)\; \; \sqrt{2x+7}\leq x+2\; \; \Leftrightarrow \; \; \left\{\begin{array}{lll}2x+7\geq 0\\x+2\geq 0\\2x+7\leq x^2+4x+4\end{array}\right\; \left\{\begin{array}{ll}x\geq -3,5\\x\geq -2\\x^2+2x-3\geq 0\end{array}\right\\\\\\\left\{\begin{array}{ll}x\geq -2\\(x-1)(x+3)\geq 0\end{array}\right\; \; \left\{\begin{array}{l}x\geq -2\\x\in (-\infty ,-3\, ]\cup [\, 1,+\infty )\end{array}\right \; \Rightarrow \; \underline {x\in [\, 1,+\infty )}$

$3)\; \; 2^{x}\cdot 5^{x-1}<0,2\cdot 10^{x}\cdot 2^{-x}\\\\2^{x}\cdot 5^{x}\cdot \frac{1}{5}<\frac{1}{5}\cdot 5^{x}\cdot 2^{x}\cdot 2^{-x}\; \; ,\; \; \; \Big (2^{x}>0\; ,\; 5^{x}>0\Big )\\\\1<2^{-x}\; \; \to \; \; \; 2^0<2^{-x}\; \; ,\; \; 2^{-x}>2^0\; ,\; \; -x>0\; ,\\\\\underline {x<0}$