Question

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Answer 1

 $9(1+5^{1-2x})^{-frac{1}{2}}-frac{1}{2}(5^{2x}+5)^{frac{1}{2}} geq 6^{frac{1}{2}}*5^{frac{x}{2}}\\ frac{9}{sqrt{1+frac{5}{5^{2x}}}} - frac{sqrt{5^{2x}+5}}{2} geq sqrt{ 6*5^x}\\ 5^{1-2x} neq -1\\ xin(-infty; + infty)\\ 5^x=t>0\\ frac{9t}{sqrt{t^2+5}}-frac{sqrt{t^2+5}}{2} geq sqrt{6t}\\ 18t-(t^2+5) geq 2sqrt{6t^3+30t}\\ -t^2+18t-5 geq 2sqrt{6t^3+30t} \\ (-t^2+18t-5)^2 -4(6t^3+30t) geq 0$    
   $(t-1)(t-5)(t^2-54t+5) geq 0\ 5^x=1\ 5^x=5\ x=0\ x=1\\ xin[0;1]$