Zad. 2.78 a) [latex]frac{2 - 4sqrt{2}}{2} =frac{2 cdot (1 - 2sqrt{2})}{2} = 1 - 2sqrt{2}[/latex] b) [latex]frac{3 - 12sqrt{5}}{-6} =frac{-3 cdot (-1+ 4sqrt{5})}{-6} =frac{-1+4sqrt{5}}{2} =frac{4sqrt{5}-1}{2} = frac{4sqrt{5}}{2} -frac{1}{2} =2sqrt{5}-frac{1}{2}[/latex] c) [latex]frac{-24 +8sqrt{2}}{-8} =frac{-8 cdot (3 -sqrt{2})}{-8} =3 -sqrt{2}[/latex] d) [latex]frac{-6 +12sqrt{3}}{24} =frac{6 cdot (-1 +2sqrt{3}}{24} =frac{-1 +2sqrt{3}}{4} =frac{2sqrt{3}-1}{4}=frac{2sqrt{3}}{4}-frac{1}{4}=frac{sqrt{3}}{2}-frac{1}{4}[/latex] e) [latex]-frac{3 -6sqrt{3}}{3} =frac{3 -6sqrt{3}}{-3} =frac{-3 cdot (-1+2sqrt{3})}{3} =-1+2sqrt{3} =2sqrt{3}-1[/latex] f) [latex]-frac{4+20sqrt{2}}{4} =frac{4+20sqrt{2}}{-4} =-frac{-4 cdot (-1-5sqrt{2})}{-4} =-1-5sqrt{2}[/latex] g) [latex]frac{-4-12pi}{12pi + 4} =frac{-1 cdot (4+12pi)}{12pi + 4} =-1[/latex] h) [latex]frac{pi-pi^2}{1-pi} =frac{pi cdot (1-pi)}{1-pi} =pi[/latex] i) [latex]frac{3 -sqrt{3}}{2sqrt{3} -6} =frac{3 -sqrt{3}}{-2 cdot (-sqrt{3}+3)} =frac{3 -sqrt{3}}{-2 cdot (3-sqrt{3})} =frac{1}{-2}=-frac{1}{2}[/latex] Zad. 2.84 a) [latex]frac{x}{3,8}= frac{1,2}{1,9} \ 1,9x = 3,8 cdot 1,2 \ 1,9x = 4,56 /:1,9 \ x =2,4[/latex] b) [latex]frac{y+3}{1,5} =frac{1frac{2}{3}}{4} \ 4 cdot (y+3)= 1,5 cdot 1frac{2}{3} \ 4y + 12 = frac{3}{2} cdot frac{5}{3} \ 4y + 12 =frac{5}{2} \ 4y = 2,5 - 12 \ 4y = - 9,5 /:4 \ y = - 2,375[/latex] c) [latex]frac{2,5}{2z-0,4}= 1frac{1}{4}[/latex] Zał. [latex]2z-0,4 eq 0 \ 2z eq 0,4 /: 2 \ z eq 0,2[/latex] [latex]frac{2,5}{2z-0,4}= 1frac{1}{4} \ frac{2,5}{2z-0,4}= frac{5}{4} \ 5 cdot (2z - 0,4) = 2,5 cdot 4 \ 5 cdot (2z - 0,4) = 10 /:5 \ 2z - 0,4 = 2 \ 2z = 2 + 0,4 \ 2z = 2,4 /:2 \ z =1,2[/latex] d) [latex]frac{2}{3} =frac{3a-4}{a +2}[/latex] Zał. [latex]a +2 eq 0 \ a eq - 2[/latex] [latex]frac{2}{3} =frac{3a-4}{a +2} \ 3 cdot (3a - 4)=2 cdot (a+2) \ 9a - 12 = 2a +4 \ 9a - 2a = 4 + 12 \ 7a = 16 /:7 \ a=frac{16}{7} \ a =2frac{2}{7}[/latex] e) [latex]frac{1frac{1}{3}b-4}{b+2,4} =frac{3}{0,75}[/latex] Zał. [latex]b+2,4 eq 0 \ b eq -2,4[/latex] [latex]frac{1frac{1}{3}b-4}{b+2,4} =frac{3}{0,75} \ frac{frac{4}{3}b-4}{b+2,4} =frac{3}{ frac{3}{4}} \ 3 cdot (b+2,4)=frac{3}{4} cdot (frac{4}{3}b-4) \ 3b+7,2=b-3 \ 3b-b =-3-7,2 \ 2b = -10,2 /:2 \ b = -5,1[/latex] f) [latex]frac{h+4}{2h} = frac{h+0,25}{2h-3}[/latex] Zał. [latex]2h eq 0 /:2 wedge 2h-3 eq 0 \ h eq 0 wedge 2h eq 3 /:2 \ h eq 0 wedge h eq 1,5[/latex] [latex]frac{h+4}{2h} = frac{h+0,25}{2h-3} \ (h+4)(2h-3) = 2h cdot (h +0,25) \ 2h^2 -3h +8h - 12 = 2h^2 +0,5h \ 2n^2+5h - 12 = 2h^2 +0,5h \ 2n^2+5h -2h^2-0,5h = 12 \ 4,5h = 12 /:4,5 \ h = 12 : 4frac{1}{2} \ h = 12 : frac{9}{2} \ h = 12 cdot frac{2}{9} \ h= frac{8}{3} \ h=2frac{2}{3}[/latex] Zad. 2.93 a) [latex]frac{3x-4}{2}= frac{4x-5}{3} \ 3 cdot (3x-4) = 2 cdot (4x-5) \ 9x - 12 = 8x - 10 \ 9x - 8x = -10+12 \ x = 2[/latex] b) [latex]frac{x}{2} = frac{8}{x}[/latex] Zał. [latex]x eq 0[/latex] [latex]frac{x}{2} = frac{8}{x} \ x^2 = 16 \ x = 4 vee x = - 4[/latex] c) [latex]frac{2}{x^2} = frac{1}{3}[/latex] Zał. [latex]x^2 eq 0 \ x eq 0[/latex] [latex]frac{2}{x^2} = frac{1}{3} \ x^2 = 6 \ x = sqrt{6} vee x = - sqrt{6}[/latex] d) [latex]frac{1}{x-1} +2 = 0[/latex] Zał. [latex] x - 1 eq 0 \ x eq 1[/latex] [latex]frac{1}{x-1} +2 = 0 / cdot (x-1) \ 1 + 2 cdot(x - 1) = 0 \ 1+2x - 2 = 0 \ 2x - 1 = 0 \ 2x = 1 /: 2 \ x = frac{1}{2}[/latex] e) [latex]frac{3 cdot (x-1)}{5} - frac{x-3}{2} =frac{x- 8}{10} / cdot 10 \ 6 cdot (x-1)- 5 cdot (x-3) = x - 10 \ 6x - 6 - 5x + 15 = x - 10 \ x +9 = x - 10 \ x - x = - 10 - 9 \ 0 = - 19 \ sprzeczno's'c[/latex] Równanie nie ma rozwiązań f) [latex]frac{3 cdot (x-11)}{4} = frac{3 cdot (x+1)}{5} -frac{2 cdot(2x- 5)}{11} / cdot 220 \ 165 cdot (x-11)=132cdot (x+1) -40cdot(2x- 5)\ 165x - 1815 =132x+132 - 80x +200 \ 165x - 1815 =52x+332 \ 165-52x = 332+1815 \ 113x =2147 /: 113 \ x =19[/latex] g) [latex]frac{x cdot (x-1)}{4}- frac{2x^2 +1}{2}= - frac{3}{4}x^2 -0,25 cdot(x+2) \ frac{x cdot (x-1)}{4}- frac{2x^2 +1}{2}= - frac{3}{4}x^2 -frac{1}{4}cdot(x+2) / cdot 4 \ x cdot (x-1)- 2 cdot (2x^2 +1)= - 3x^2 -(x+2) \ x^2 - x -4x^2-2 = - 3x^2 -x - 2 \ -3x^2 -x - 2 = - 3x^2 -x - 2 \ - 3x^2 -x+3x^2+x= - 2+2 \ 0 = 0 \ to.zsamo's'c [/latex] Równanie ma nieskończenie wiele rozwiązań, czyli [latex]x in R[/latex] h) [latex]frac{x+ frac{4-x}{3}}{5} = frac{x- frac{x-5}{5}}{3} / cdot 15 \ 3 cdot (x+ frac{4-x}{3})=5 cdot(x- frac{x-5}{5}) \ 3x+(4-x)=5x- (x-5) \ 3x+4-x = 5x-x+5 \ 2x+4 =4x+5 \ 2x-4x = 5 - 4 \ -2x = 1 /:(-2) \ x = -frac{1}{2}[/latex] Zad. 2.95 a) [latex]sqrt{2}x - 5 = 0 \ sqrt{2}x = 5 /:sqrt{2} \ x = frac{5}{sqrt{2}} \ x = frac{5 cdot sqrt{2}}{sqrt{2} cdot sqrt{2}} \ x = frac{5sqrt{2}}{2} \ x =2,5sqrt{2}[/latex] b) [latex]1 +sqrt{3}x = x+2 \ sqrt{3}x -x = 2 - 1 \ (sqrt{3}-1) cdot x = 1 /:(sqrt{3}-1) \ x = frac{1}{sqrt{3}-1} \ x = frac{sqrt{3}+1}{(sqrt{3}-1)(sqrt{3}+1)} \ x = frac{sqrt{3}+1}{3-1} \ x= frac{sqrt{3}+1}{2}[/latex] c) [latex]2 - sqrt{2}x = x-4 \ -sqrt{2}x-x=-4- 2 \ -sqrt{2}x-x=-6 / cdot(-1) \ sqrt{2}x+x=6 \ (sqrt{2} +1) cdot x = 6 / :(sqrt{2} +1) \ x = frac{6}{sqrt{2} +1} \ x = frac{6 cdot (sqrt{2}-1)}{(sqrt{2} +1)(sqrt{2}-1)} \ x = frac{6 cdot (sqrt{2}-1)}{2-1} \ x = frac{6 cdot (sqrt{2}-1)}{1} \ x =6 cdot (sqrt{2}-1)[/latex] d) [latex]2x+3+xsqrt{3} = 4 \ 2x+xsqrt{3} = 4 - 3 \ (2 + sqrt{3}) cdot x = 1 /:(2 + sqrt{3}) \ x = frac{1}{2 + sqrt{3}} \ x =frac{2-sqrt{3}}{(2+sqrt{3})(2-sqrt{3})} \ x = frac{2-sqrt{3}}{4-3} \ x = frac{2-sqrt{3}}{1} \ x =2-sqrt{3}[/latex] e) [latex](x+2sqrt{2}) cdot sqrt{2} = sqrt{3}x-1 \ sqrt{2}x+4= sqrt{3}x-1 \ sqrt{2}x-sqrt{3}x=-1-4 \ (sqrt{2}-sqrt{3}) cdot x=-5 /:(sqrt{2}-sqrt{3}) \ x = frac{-5}{sqrt{2}-sqrt{3}} \ x =frac{-5}{-(sqrt{3}-sqrt{2})} \ x =frac{5}{sqrt{3}-sqrt{2}} \ x =frac{5 cdot (sqrt{3}+sqrt{2})}{(sqrt{3}-sqrt{2})(sqrt{3}+sqrt{2})} \ x =frac{5 cdot (sqrt{3}+sqrt{2})}{3-2} \ x= frac{5 cdot (sqrt{3}+sqrt{2})}{1} \ x=5 cdot (sqrt{3}+sqrt{2})[/latex] f) [latex]sqrt{5}(x-sqrt{5}) = 10 -x \ sqrt{5}x - 5 = 10 - x \ sqrt{5}x - x = 10+5 (sqrt{5} - 1) cdot x = 15 / :(sqrt{5}-1) \ x = frac{15}{sqrt{5}-1} \ x =frac{15 cdot (sqrt{5}+1))}{(sqrt{5}-1)(sqrt{5}+1)} \ x = frac{15 cdot(sqrt{5}+1)}{5-1} \ x =frac{15 cdot(sqrt{5}+1)}{4}[/latex] g) [latex]frac{1}{sqrt{3} +x} = frac{sqrt{3}}{x}[/latex] Zał. [latex]sqrt{3} +x eq 0 wedge x eq 0 \ x eq -sqrt{3} wedge x eq 0[/latex] [latex]frac{1}{sqrt{3} +x} = frac{sqrt{3}}{x} \ sqrt{3} cdot (sqrt{3}+x} = x \ 3 + sqrt{3}x = x \ sqrt{3}x -x= -3 \ (sqrt{3}-1) cdot x = - 3 /:(sqrt{3}-1) \ x = frac{-3 }{sqrt{3}-1} \ x =- frac{3 cdot(sqrt{3}+1)}{(sqrt{3}-1)(sqrt{3}+1)} \ x = - frac{3 cdot(sqrt{3}+1)}{3-1} \ x= - frac{3 cdot(sqrt{3}+1)}{2}[/latex] h) [latex]frac{2x-3}{xsqrt{7}} = frac{1}{2}[/latex] Zał. [latex]xsqrt{7} eq 0 /:sqrt{7} \ x eq 0[/latex] [latex]frac{2x-3}{xsqrt{7}} = frac{1}{2} \ 2 cdot (2x - 3)=xsqrt{7}\ 4x - 6=xsqrt{7}\ 4x - xsqrt{7} = 6 \ (4-sqrt{7}) cdot x =6 / :(4-sqrt{7}) \ x = frac{6}{4-sqrt{7}} \ x = frac{6 cdot (4+sqrt{7})}{(4-sqrt{7})(4+sqrt{7})} x=frac{6 cdot (4+sqrt{7})}{16-7} \ x=frac{6 cdot (4+sqrt{7})}{9} \ x=frac{2 cdot (4+sqrt{7})}{3}[/latex]
