Zad. 3
a)
[tex](2-3x)^{2} \geq 9x^{2} \\4-12x+9x^{2} \geq 9x^{2} \\4-12x\geq 0\\-12x\geq -4\\x\leq \frac{1}{3}[/tex]
b)
[tex](3-2x)^{2}-4(1-x)^{2}<0\\9-12x+4x^{2}-4(1-2x+x^{2})<0\\9-12x+4x^{2} -4+8x-4x^{2}<0\\5-4x<0\\-4x<-5\\x>\frac{5}{4}[/tex]
c)
[tex](1+4x)*(4x-1)\geq (1-4x)^{2} \\(4x+1)*(4x-1)\geq 1-8x+16x^{2} \\16x^{2} - 1\geq 1-8x+16x^{2} \\-1\geq 1-8x\\8x\geq 1+1\\8x\geq 2\\x\geq \frac{1}{4} \\[/tex]
d)
[tex](5x+3)^{2} -(4x+3)^{2} >(3x+3)^{2} \\25x^{2} +30x+9-(16x^{2} +24x+9)>9x^{2} +18x+9\\25x^{2} +30x-16x^{2} -24x-9>9x^{2} +18x\\9x^{2} +6x-9>9x^{2} +18x\\6x-9>18x\\6x-18x>9\\-12x>9\\x<-\frac{3}{4} \\[/tex]
Zad. 4
a)
[tex]\frac{1}{\sqrt{5}+1 }\\\\\frac{1}{\sqrt{5}+1 } * \frac{\sqrt{5}-1 }{\sqrt{5}1 } \\\frac{1(\sqrt{5}-1) }{(\sqrt{5} +1) } \\\frac{\sqrt{5}-1 }{5-1 } \\\frac{\sqrt{5}-1 }{4}[/tex]
b)
[tex]\frac{7}{\sqrt{8} +1} \\\frac{7}{2\sqrt{2} +1} \\\frac{7}{2\sqrt{2}+1 } * \frac{2\sqrt{2}-1 }{2\sqrt{2}-1 } \\\frac{7(2\sqrt{2}-1) }{(2\sqrt{2}+1)*(2\sqrt{2}-1) } \\\frac{7(2\sqrt{2}-1) }{4*2-1} \\\frac{7(2\sqrt{2}-1) }{8-1} \\\frac{7(2\sqrt{2}-1) }{7} \\2\sqrt{2} - 1\\[/tex]
c)
[tex]\frac{8}{\sqrt{6}-2 } \\\frac{8}{\sqrt{6}-2 } * \frac{\sqrt{6}+2 }{\sqrt{6}+2 } \\\frac{8(\sqrt{6}+2) }{(\sqrt{6}-2)*(\sqrt{6}+2) } \\\frac{8(\sqrt{6}+2) }{6-4} \\\frac{8(\sqrt{6}+2) }{2} \\4(\sqrt{6} +2)\\4\sqrt{6} +8\\[/tex]
d)
[tex]\frac{\sqrt{3} }{\sqrt{6}-2 }\\\frac{\sqrt{3} }{\sqrt{6}-2 }*\frac{\sqrt{6}+2 }{\sqrt{6}+2 } \\\frac{\sqrt{3}(\sqrt{6}+2) }{(\sqrt{6}-2)*(\sqrt{6}+2) } \\\frac{\sqrt{18}+2\sqrt{3} }{6-4} \\\frac{3\sqrt{2}+2\sqrt{3} }{2}\\[/tex]
e)
[tex]\frac{\sqrt{8} }{4+\sqrt{2} } \\\frac{2\sqrt{2} }{4+\sqrt{2} } \\\frac{2\sqrt{2} }{4+\sqrt{2} }*\frac{4-\sqrt{2} }{4-\sqrt{2} } \\\frac{2\sqrt{2}(4-\sqrt{2}) }{(4+\sqrt{2})*(4-\sqrt{2} } \\\frac{2\sqrt{2}(4-\sqrt{2}) }{16-2} \\\frac{2\sqrt{2}(4-\sqrt{2}) }{14} \\\frac{\sqrt{2}(4-\sqrt{2}) }{7} \\\frac{4\sqrt{2}-2 }{7} \\[/tex]
f)
[tex]\frac{4+\sqrt{5}}{4-\sqrt{5}} \\\frac{4+\sqrt{5}}{4-\sqrt{5}}*\frac{4+\sqrt{5}}{4+\sqrt{5}} \\\frac{(4+\sqrt{5})*(4+\sqrt{5})}{(4-\sqrt{5})*(4+\sqrt{5})} \\\frac{(4+\sqrt{5})^{2}}{16-5} \\\frac{16+8\sqrt{5}+5}{11} \\\frac{21+8\sqrt{5}}{11} \\[/tex]
