Question

3^(2*(2x+5))*5^(2*(3x+1))-15^(5x+6)<0

Answer 1

$3^{2(2x+5)}\cdot 5^{2(3x+1)}-15^{5x+6}<0\\\\3^{4x}\cdot 3^{10}\cdot 5^{6x}\cdot 5^{2}-(3\cdot 5)^{5x+6}<0\\\\3^{10}\cdot 5^2\cdot 3^{4x}\cdot 5^{6x}-3^{5x}\cdot 3^6\cdot 5^{5x}\cdot 5^6<0\\\\3^6\cdot 5^2\cdot 3^{4x}\cdot 5^{5x}\cdot \Big (3^4\cdot 5^{x}-5^4\cdot 3^{x}\Big )<0\\\\3^6\cdot 5^2\cdot 3^{4x}\cdot 5^{5x}>0\; \; \Rightarrow \; \; 3^4\cdot 5^{x}-5^4\cdot 3^{x}<0\\\\3^4\cdot 5^{x}<5^4\cdot 3^{x}\, \Big |:(3^4\cdot 3^{x})>0\\\\\frac{5^{x}}{3^{x}}<\frac{5^4}{3^4}\\\\(\frac{5}{3})^{x}<(\frac{5}{3})^4\; \; \; \Rightarrow \; \; x<4\\\\x\in (-\infty ,4)$