Zadanie 1. a) [latex](x-sqrt2)(x+2sqrt2)=x^2+1\ xcdot x+xcdot 2sqrt2-sqrt2cdot x-sqrt2cdot 2sqrt2=x^2+1\ x^2+2sqrt2x-sqrt2x-4=x^2+1\ x^2+sqrt2x-4=x^2+1\ x^2+sqrt2x-4-x^2=1\ sqrt2x=1+4\ sqrt2x=5 /:sqrt2\ x= frac{5}{sqrt2} cdot frac{sqrt2}{sqrt2} \ x = frac{5sqrt2}{2} [/latex] b) [latex]1+sqrt3x=x+2\ sqrt3x-x=2-1\ x(sqrt3-1)=1/:(sqrt3-1)\ x = frac{1}{sqrt3-1} cdot frac{sqrt3+1}{sqrt3+1} \ x = frac{sqrt3+1}{(sqrt3)^2 -1^2}\ x = frac{sqrt3+1}{3-1} \ x= frac{sqrt3+1}{2} [/latex] c) [latex]2-sqrt2x=x-4\ -sqrt2-x=-4-2 /cdot (-1)\ sqrt2x+x=4+2\ x(sqrt2+1)=6 /:(sqrt2+1) \ x= frac{6}{sqrt2+1} cdot frac{sqrt2-1}{sqrt2-1} \ x = frac{6(sqrt2-1)}{(sqrt2)^2-1^2}\ x = frac{6(sqrt2-1)}{2-1}\ x = frac{6sqrt2-6}{1}\ x = 6sqrt2 -6 [/latex] d) [latex](x+sqrt3)(2+sqrt3)=4+2sqrt3 /:(2+sqrt3)\ x+sqrt3 = frac{4+2sqrt3 }{2+sqrt3} cdot frac{2-sqrt3}{2-sqrt3} \ x+sqrt3 = frac{(4+2sqrt3 )(2-sqrt3)}{2^2 -(sqrt3)^2} \ x+sqrt3 = frac{8-4sqrt3+4sqrt3-6}{4-3} \ x+sqrt3 = frac{2}{1} \ x+sqrt3 = 2\ x=2-sqrt3[/latex] e) [latex](x+2sqrt2)sqrt2=sqrt3x-1\ sqrt2x+4=sqrt3x-1\ sqrt2x-sqrt3x = -1-4\ x(sqrt2-sqrt3) = -5/:(sqrt2-sqrt3)\ x = frac{-5}{sqrt2-sqrt3} cdot frac{sqrt2+sqrt3}{sqrt2+sqrt3} \ x = frac{-5(sqrt2+sqrt3)}{(sqrt2)^2-(sqrt3)^2}\ x = frac{-5(sqrt2+sqrt3)}{2-3}\ x = frac{-5(sqrt2+sqrt3)}{-1}\ x = 5(sqrt2+sqrt3)\ x = 5sqrt2+5sqrt3[/latex] f) [latex](x+sqrt5)(1+sqrt5)=sqrt5 /:(1+sqrt5)\ x+sqrt5 = frac{sqrt5}{1+sqrt5} cdot frac{1-sqrt5}{1-sqrt5} \ x+sqrt5 = frac{sqrt5(1-sqrt5)}{1^2 - (sqrt5)^2} \ x+sqrt5 = frac{sqrt5 -5}{1-5} \ x+sqrt5 = frac{sqrt5 -5}{-4}\ x+sqrt5 = frac{5-sqrt5}{4}\ x = frac{5-sqrt5}{4} -sqrt5\ x = frac{5-sqrt5}{4}- frac{4sqrt5}{4}\ x = frac{5-sqrt5-4sqrt5}{4}\ x = frac{5-5sqrt5}{4}[/latex] Zadanie 2. a) 2x-5>2(x+3) 2x-5>2x+6 2x-2x>6+5 0>11 sprzeczność b) 4,2(x+5)+1,8[latex] leq [/latex]-2,4 4,2x+21+1,8[latex] leq [/latex] -2,4 4,2x[latex] leq [/latex]-2,4-22,8 4,2x [latex] leq [/latex] 25,2 /:4,2 x[latex] leq [/latex]6[latex] x in (-infty,6>[/latex] c) [latex](x-2)(2x+5)>3x+4+2x^2 \ 2x^2+5x-4x-10>3x+4+2x^2\ 2x^2+x-3x-2x^2>4+10\ -2x>14/:(-2)\ x<-7\ x in (- infty,-7)[/latex] d) [latex] frac{5,6x-4}{8} leq 0,3x+4,5 /cdot 8\ 5,6x-4 leq 2,4x+ 36\ 5,6x-2,4x leq 36+4\ 3,2x leq 40/:3,2\ x leq 12,5\ x in(- infty;12,5>[/latex] e) [latex] frac{x+2}{3}- frac{x+4}{4} > frac{x+1}{12}/cdot 12\ 4(x+2)-3(x+4)>x+1\ 4x+8-3x-12>x+1\ x-4>x+1\ x-x>1+4\ 0>5\ sprzecznosc [/latex] f) [latex]-1 frac{2}{3} (x+6)> frac{1}{6} (x+3)+0,5\ - frac{5}{3} (x+6)>frac{1}{6} (x+3)+ frac{1}{2} /cdot 6\ -10(x+6)>x+3+3\ -10x-60>x+6\ -10x-x>6+60\ -11x>66/:(-11)\ x<-6\ x in (-infty,-6)[/latex] g) [latex] frac{x-5}{7} < frac{3x+2}{21}/cdot 21\ 3(x-5)<3x+2\ 3x-15>3x+2\ 3x-3x>2+15\ 0>17 \ sprzecznosc [/latex] h) [latex](3-x)(x+1) geq frac{5-2x^2}{2} /cdot 2\ 2(3x+3-x^2-x) geq 5-2x^2\ 6x+6-2x^2-2x geq 5-2x^2\ -2x^2+4x+6 geq 5-2x^2\ -2x^2+4x+2x^2 geq 5-6\ 4x geq -1 /:4\ x geq - frac{1}{4} \ x in <- frac{1}{4} , infty)[/latex] Zadanie 3. a) [latex]2(x-1)-3(x-2)<6\ 2x-2-3x+6<6\ -x+4<6\ -x<6-4\ -x<2 /cdot 9-1)\ x>-2\ x in(-2, infty)[/latex] b) [latex]-3(2x+4)+5(x-2) geq -x+5 \ -6x-12+5x-10 geq -x+5\ -x-22 geq -x+5\ -x+x geq 5+22\ 0 geq 27\ sprzecznosc[/latex] c) [latex] frac{4-x}{5} - frac{x+2}{3} geq frac{2-8x}{15}/cdot 15\ 3(4-x)-5(x+2) geq 2-8x \ 12+3x-5x-10 geq 2-8x\ -2x+2 geq 2-8x\ -2x+8x geq 2-2\ 6x geq 0 /:6\ x geq 0\ x in <0 , infty) [/latex] d) [latex] frac{x-5}{2}- frac{3-x}{6} leq frac{x+4}{3} /cdot 6\ 3(x-5)-(3-x) leq 2(x+4)\ 3x-15-3+x leq 2x+8\ 4x-18 leq 2x+8\ 4x-2x leq 8+18\ 2x leq 26 /:2\ x leq 13\ x in (-infty, 13>[/latex] e) [latex] frac{x- frac{3-2x}{2} }{5} >2 /cdot 5\ x- frac{3-2x}{2} >10 /cdot 2 \ 2x-(3-2x)>20 \ 2x-3+2x>20 \ 4x>20+3 \ 4x>23 /:4 \ x> frac{23}{4}\ x>5 frac{3}{4} \ x in (5 frac{3}{4} , infty)[/latex] f) [latex] frac{ frac{x-2,6}{3} -2x}{4} geq frac{x+2}{5} /cdot 20\ 5(frac{x-2,6}{3} -2x) geq 4(x+2)\ frac{5(x-2,6)}{3} -10x geq 4x+8 / cdot 3 \ 5x-13-10x geq 12x+24\ -5x-13 geq 12x+24\ -5x-12x geq 24+13\ -17x geq 37 /:(-17)\ x leq - frac{37}{17} \ x leq -2 frac{3}{17} \ x in (-infty,-2 frac{3}{17} >[/latex]
