4^{x}+4^{x+1}\\\\3^{x}\cdot (3+3^2+3^3)>4^{x}\cdot (1+4)\\\\3^{x}\cdot 39>4^{x}\cdot 5\; \; \Rightarrow \; \; \frac{3^{x}}{4^{x}}>\frac{5}{39}\; \; ,\; \; (\frac{3}{4})^{x}>\frac{5}{39}\; \; \Rightarrow \; \; x0\\\\(3^{2x^2-5x})^2+3^{2x^2-5x}\cdot 3-4>0\\\\t=3^{2x^2-5x}>0\; \; ,\; \; t^2+3t-4>0\; ,\; \; t_1=-4\; ,\; t_2=1\; \; (Viet)\\\\znaki:\; \; \; +++(-4)---(1)+++\; \; \;\qquad t>1\; \; ili\; \; t<-4" alt="1)\; \; 3^{x+1}+3^{x+2}+3^{x+3}>4^{x}+4^{x+1}\\\\3^{x}\cdot (3+3^2+3^3)>4^{x}\cdot (1+4)\\\\3^{x}\cdot 39>4^{x}\cdot 5\; \; \Rightarrow \; \; \frac{3^{x}}{4^{x}}>\frac{5}{39}\; \; ,\; \; (\frac{3}{4})^{x}>\frac{5}{39}\; \; \Rightarrow \; \; x0\\\\(3^{2x^2-5x})^2+3^{2x^2-5x}\cdot 3-4>0\\\\t=3^{2x^2-5x}>0\; \; ,\; \; t^2+3t-4>0\; ,\; \; t_1=-4\; ,\; t_2=1\; \; (Viet)\\\\znaki:\; \; \; +++(-4)---(1)+++\; \; \;\qquad t>1\; \; ili\; \; t<-4" align="absmiddle" class="latex-formula">
1\; \; ,\; \; 3^{2x^2-5x}>3^0\; \; \to \; \; 2x^2-5x>0\; ,\\\\x(2x-5)>0\; \; ,\; \; x_1=0\; ,\; x_2=2,5\\\\+++(0)---(2,5)+++\\\\x\in (-\infty ;0)\cup (2,5\, ;\, +\infty )\\\\b)\; \; 3^{2x^2-5x}<-4\; \; \Rightarrow \; \; x\in \varnothing\; ,\; t.k.\; \; 3^{2x^2-5x}>0\; .\\\\Otvet:\; \; x\in (-\infty \, ;0)\cup (2,5\, ;+\infty )\; ." alt="a)\; \; 3^{2x^2-5x}>1\; \; ,\; \; 3^{2x^2-5x}>3^0\; \; \to \; \; 2x^2-5x>0\; ,\\\\x(2x-5)>0\; \; ,\; \; x_1=0\; ,\; x_2=2,5\\\\+++(0)---(2,5)+++\\\\x\in (-\infty ;0)\cup (2,5\, ;\, +\infty )\\\\b)\; \; 3^{2x^2-5x}<-4\; \; \Rightarrow \; \; x\in \varnothing\; ,\; t.k.\; \; 3^{2x^2-5x}>0\; .\\\\Otvet:\; \; x\in (-\infty \, ;0)\cup (2,5\, ;+\infty )\; ." align="absmiddle" class="latex-formula">
0\; \; ,\; \; \; 3t+\frac{2}{t}-5\leq 0\; ,\; \; \; \frac{3t^2-5t+2}{t}\leq 0\; ,\; \; 3t^2-5t+2\leq 0\; ,\\\\D=1\; ,\; t_1=\frac{2}{3}\; \; ,\; \; t_2=1\\\\znaki:\; \; \; +++[\, \frac{2}{3}\, ]---[\, 1\, ]+++\; \; \; \quad t\in [\, \frac{2}{3}\, ;\, 1\, ]" alt="3)\; \; 3\cdot 16^{x}+2\cdot 81^{x}\leq 5\cdot 36^{x}\\\\3\cdot (2^{x})^4+2\cdot (3^{x})^4-5\cdot (6^{x})^2\leq 0\; |:(6^{x})^2\\\\3\cdot (\frac{2^{x}}{3^{x}})^2+2\xsot (\frac{3^{x}}{2^{x}})^2-5\leq 0\\\\t=(\frac{2^{x}}{3^{x}})^2>0\; \; ,\; \; \; 3t+\frac{2}{t}-5\leq 0\; ,\; \; \; \frac{3t^2-5t+2}{t}\leq 0\; ,\; \; 3t^2-5t+2\leq 0\; ,\\\\D=1\; ,\; t_1=\frac{2}{3}\; \; ,\; \; t_2=1\\\\znaki:\; \; \; +++[\, \frac{2}{3}\, ]---[\, 1\, ]+++\; \; \; \quad t\in [\, \frac{2}{3}\, ;\, 1\, ]" align="absmiddle" class="latex-formula">
![(\frac{2^{x}}{3^{x}})^{2}=(\frac{2}{3})^{2x}\; \; ,\; \; \; \; \frac{2}{3}\leq (\frac{2}{3})^{2x}\leq 1\; \; \Rightarrow \; \; 0\leq 2x\leq 1\; \; ,\; \; 0\leq x\leq 0,5\\\\Otvet:\; \; x\in [\, 0\, ;\, 0,5\, ]\; .](https://tex.z-dn.net/?f=%28%5Cfrac%7B2%5E%7Bx%7D%7D%7B3%5E%7Bx%7D%7D%29%5E%7B2%7D%3D%28%5Cfrac%7B2%7D%7B3%7D%29%5E%7B2x%7D%5C%3B%20%5C%3B%20%2C%5C%3B%20%5C%3B%20%5C%3B%20%5C%3B%20%5Cfrac%7B2%7D%7B3%7D%5Cleq%20%28%5Cfrac%7B2%7D%7B3%7D%29%5E%7B2x%7D%5Cleq%201%5C%3B%20%5C%3B%20%5CRightarrow%20%5C%3B%20%5C%3B%200%5Cleq%202x%5Cleq%201%5C%3B%20%5C%3B%20%2C%5C%3B%20%5C%3B%200%5Cleq%20x%5Cleq%200%2C5%5C%5C%5C%5COtvet%3A%5C%3B%20%5C%3B%20x%5Cin%20%5B%5C%2C%200%5C%2C%20%3B%5C%2C%200%2C5%5C%2C%20%5D%5C%3B%20. "(\frac{2^{x}}{3^{x}})^{2}=(\frac{2}{3})^{2x}\; \; ,\; \; \; \; \frac{2}{3}\leq (\frac{2}{3})^{2x}\leq 1\; \; \Rightarrow \; \; 0\leq 2x\leq 1\; \; ,\; \; 0\leq x\leq 0,5\\\\Otvet:\; \; x\in [\, 0\, ;\, 0,5\, ]\; .")
1\\\\(x^2-6x+8)\cdot lg(x-2)>lg1\qquad [\; x^2-6x+8=0\; \to \; x_1=2\; ,\; x_2=4\, ]\\\\(x^2-6x+8)\cdot lg(x-2)>0\\\\\left \{ {{x^2-6x+8>0} \atop {lg(x-2)>0}} \right. \; \; ili\; \; \left \{ {{x^2-6x+8<0} \atop {lg(x-2)<0}} \right. \\\\\left \{ {{(x-2)(x-4)>0} \atop {x-2>1}} \right. \; \; ili\; \; \left \{ {{(x-2)(x-4)<0} \atop {x-2<1}} \right.\\\\\left \{ {{x\in (-\infty ,2)\cup (4,+\infty )} \atop {x>3}} \right.\; \; ili\; \; \left \{ {{x\in (2,4)} \atop {x<3}} \right." alt="4)\; \; (x-2)^{x^2-6x+8}>1\\\\(x^2-6x+8)\cdot lg(x-2)>lg1\qquad [\; x^2-6x+8=0\; \to \; x_1=2\; ,\; x_2=4\, ]\\\\(x^2-6x+8)\cdot lg(x-2)>0\\\\\left \{ {{x^2-6x+8>0} \atop {lg(x-2)>0}} \right. \; \; ili\; \; \left \{ {{x^2-6x+8<0} \atop {lg(x-2)<0}} \right. \\\\\left \{ {{(x-2)(x-4)>0} \atop {x-2>1}} \right. \; \; ili\; \; \left \{ {{(x-2)(x-4)<0} \atop {x-2<1}} \right.\\\\\left \{ {{x\in (-\infty ,2)\cup (4,+\infty )} \atop {x>3}} \right.\; \; ili\; \; \left \{ {{x\in (2,4)} \atop {x<3}} \right." align="absmiddle" class="latex-formula">
