[latex]a)\ x^{2} extgreater 25 \ x^2-25 extgreater 0\ (x+5)(x-5) extgreater 0\ x=-5 vee x=5\ \ xin (-infty,-5)cup(5,+infty)\ \ \ b) \ x^{2} leq 16 \ x^2-16le 0\ (x+4)(x-4)le 0\ x=-4 vee x=4\ \ xin extless -4,4 extgreater \ \ \ c)\ x^{2} - x geq 6 \ x^2-x-6ge 0\ Delta = (-1)^2-4cdot 1cdot (-6)=1+24=25\ x=frac{1-5}{2}=frac{-4}{2}=-2\ x=frac{1+5}{2}=frac{6}{2}=3\ \ xin (-infty, -2 extgreater cup extless 3,+infty) [/latex]
[latex]d)\ x^{2} - 4 leq 3x \ x^2-3x-4le 0\ Delta=(-3)^2-4cdot 1cdot (-4)=9+16=25\ x=frac{3-5}{2}=frac{-2}{2}=-1\ x=frac{3+5}{2}=frac{8}{2}=4\ \ xin extless -1,4 extgreater \ \ \ e)\ 4x+5 extgreater x^{2} \ x^2-4x-5 extless 0\ Delta = (-4)^2-4cdot 1cdot (-5)=16+20 = 36\ x=frac{4-6}{2}=frac{-2}{2}=-1\ x=frac{4+6}{2}=frac{10}{2}=5\ \ xin (-1,5)\ \ \ [/latex]
[latex]f) \ 2 x^{2} + 5x extgreater 3 \ 2x^2+5x-3 extgreater 0\ Delta = 5^2-4cdot 2cdot (-3)=25+24=49\ x=frac{-5-7}{4}=frac{-12}{4}=-3\ x=frac{-5+7}{4}=frac{2}{4}=frac{1}{2}\ \ xin(-infty, -3)cup (frac{1}{2},+infty)\ \ \ g)\ 6 x^{2} - x extgreater 12 \ 6x^2-x-12 extgreater 0\ Delta=(-1)^2-4cdot 6cdot12 = 1+288=289\ x=frac{1-17}{12}=frac{-16}{12}=-frac{4}{3}\ x=frac{1+17}{12}=frac{18}{12}=frac{3}{2}\ \ xin(-infty, -frac{4}{3})cup(frac{3}{2},+infty)\ \ \ [/latex]
[latex]h) \3x + 8 geq frac{1}{2} x^{2} \ frac{1}{2}x^2-3x-8le 0\ Delta = (-3)^2-4cdot frac{1}{2}cdot (-8)=9+16=25\ x=frac{3-5}{1}=-2\ x=frac{3+5}{1}=8\ \ xin extless -2,8 extgreater [/latex]
