[latex]1.50 \a) 0,5^{2x^2-x} geq 1 \0,5^{2x^2-x} geq 0,5^0 \2x^2-x leq 0 \x(2x-1) leq 0 \\underline{+++++}langle0 angleunderline{----}langle frac{1}{2} angle underline{++++} \\x in langle 0, frac{1}{2} angle[/latex] [latex]b)(1 frac{1}{3})^{x^2-x}<( frac{3}{4})^{-6} \( frac{4}{3})^{x^2-x}<( frac{4}{3} )^6 \x^2-x<6 \x^2-x-6<0 \Delta=1+24=25 Rightarrow sqrt{Delta}=5 \x_1= frac{1-5}{2}=-2 \x_2= frac{1+5}{2}=3 \\underline{++++}(-2)underline{---}(3)underline{++++} \\x in (-2,3) [/latex] [latex]c)3^{x^2+4}>81^x \3^{x^2+4}>(3^4)^x \3^{x^2+4}>3^4^x \x^2+4>4x \x^2-4x+4>0 \(x-2)^2>0 \x in mathbb{R}/ {2}[/latex] [latex]d)( frac{3}{7})^{x^2-4x} geq ( frac{7}{3})^3 \( frac{3}{7})^{x^2-4x} geq ( frac{3}{7})^-^3 \x^2-4x leq -3 \x^2-4x+3 leq 0 \Delta=16-12=4 Rightarrow sqrt{Delta}=2 \x_1= frac{4-2}{2}=1 \x_2= frac{4+2}{2}=3 \\underline{++++}langle1 angleunderline{---}langle3 angleunderline{++++} \\x in langle1,3 angle[/latex] [latex]e)0,5^{x^2}>2^{-4} \(2^{-1})^{x^2}>2^{-4} \2^{-x^2}>2^{-4} \-x^2>-4 \x^2<4 \(x-2)(x+2)<0 \\underline{++++}(-2)underline{----}(2)underline{++++} \\x in (-2,2)[/latex] [latex]f)0,125^x leq 2^{x^2-10} \(2^{-3})^x leq 2^{x^2-10} \2^{-3x} leq 2^{x^2-10} \-3x leq x^2-10 \-x^2-3x+10 leq 0 \x^2+3x-10 geq 0 \Delta=9+40=49 Rightarrow sqrt{Delta}=7 \x_1= frac{-3-7}{2}=-5 \x_2= frac{-3+7}{2} =2 \\underline{++++}langle-5 angleunderline{---}langle2 angleunderline{++++} \\x in (-infty,-5 angle cup langle 2, +infty) [/latex] [latex]Zad1.51 \a)0,5^{x^2-9x+17,5}< frac{8}{ sqrt{2} } \2^{-x^2+9x-17,5}<2^{ frac{5}{2} } \-x^2+9x-17,5< 2,5 \-x^2+9x-20<0 \Delta=81-80=1= sqrt{Delta} \x_1= frac{-9-1}{-2}=5 \x_2= frac{-9+1}{-2}=4 \\underline{----}(4)underline{+++}(5)underline{----} \x in (-infty,4)cup(5,+infty) [/latex] [latex]b) frac{3^{x^2}}{( sqrt{3})^{x+ frac{1}{2} } } geq sqrt[4]{3} \3^{x^2}:3^{ frac{1}{2} x+ frac{1}{4} } geq 3^{ frac{1}{4} } \3^{x^2- frac{x}{2} - frac{1}{4} }geq 3^{ frac{1}{4} } \x^2- frac{x}{2} - frac{1}{4} geq frac{1}{4} quad /*2 \2x^2-x-1 geq 0 \Delta=1+8=9 Rightarrow sqrt{Delta}=3 \x_1= frac{1-3}{4}=- frac{1}{2} \x_2= frac{1+3}{4}=1 \\underline{++++}langle- frac{1}{2} angleunderline{---}langle1 angleunderline{++++} [/latex] [latex]\\x in (-infty, - frac{1}{2} angle cup langle 1,+infty) [/latex] [latex]c) frac{8^ frac{2}{3} *2^2^x}{4^ frac{x^2}{2} } < frac{1}{64} \ 2^2*2^2^x:2^{x^2}<2^{-6} \2^{-x^2+2x+2}<2^{-6} \-x^2+2x+2<-6 \-x^2+2x+8<0 \Delta=4+32=36 Rightarrow sqrt{Delta}=6 \x_1= frac{-2-6}{-2}=4 \x_2= frac{-2+6}{-2}=-2 \\underline{----}(-2)underline{+++}(4)underline{----} \\x in (-infty,-2)cup(4,+infty)[/latex] [latex]d) frac{25^{x^2}}{( sqrt[4]{5} )^{8x}} leq frac{625}{5^{4x}} \5^{2x^2}:5^{2x} leq 5^4:5^{4x} \5^{2x^2-2x} leq 5^{4-4x} \2x^2-2x leq 4-4x \2x^2+2x-4 leq 0 \x^2+x-2 leq 0 \Delta=1+8=9 Rightarrow sqrt{Delta}=3 \x_1= frac{-1-3}{2}=-2 \x_2= frac{-1+3}{2} =1 \\underline{+++}langle-2 angleunderline{---}langle1 angleunderline{+++} \\x in langle -2,1 angle[/latex]
