1. [latex] frac{ x^{2} -3x-10}{x+2} = frac{(x+2)(x-5)}{x+2} =x-5 \ frac{x-1}{ x^{2} -1} = frac{x-1}{ (x-1)(x+1)} = frac{1}{x+1} [/latex] 2. [latex]x+1 eq 0~~~~x eq -1 \ x-3 eq 0~~~~x eq 3 \ frac{x-2}{x+1} - frac{2x}{x-3} = frac{(x-2)(x-3)}{(x+1)(x-3)} - frac{2x(x+1)}{(x-3)(x+1)} = frac{( x^{2} -2x-3x+6)-(2 x^{2} +2x)}{(x+1)(x-3)} = \ = frac{ x^{2} -5x+6-2 x^{2} -2x}{ (x+1)(x-3)} = frac{ x^{2} -5x+6-2 x^{2} -2x}{ x^{2} +x-3x-3} = frac{ -x^{2} -7x+6}{ x^{2} -2x-3} \ [/latex] [latex]x-4 eq 0~~~~x eq 4 \ frac{3x}{x-4}- frac{1}{2} = frac{3x*2}{(x-4)*2}- frac{x-4}{2(x-4)} = frac{6x-(x-4)}{2x-8} = frac{6x-x+4}{2x-8} = frac{5x+4}{2x-8} \ [/latex] [latex]x eq 0 \frac{2}{x} -3= frac{2}{x} - frac{3}{1} =frac{2}{x} - frac{3x}{x} = frac{2-3x}{x} [/latex] 3. [latex]frac{1-4x}{2x+1} =3 \ 3(2x+1)=1-4x \ 6x+3=1-4x \ 6x+4x=1-3 \ 10x=-2 \ x= -frac{2}{10} \ x=-0.2 \ \ frac{2-x-3}{x+1}=2 \x+1 eq 0 ~~~x eq -1 \ 2(x+1) =2 x^{2} -x-3 \ 2x+2=2 x^{2} -x-3 \ 2 x^{2} -x-2x-3-2=0 \[/latex] [latex] 2 x^{2} -3x-5=0 \ з=(-3)^2-4*(-5)*2=9+40=49=7^2 \ x_1= frac{-(-3)- sqrt{49} }{2*2} = frac{3-7}{4} = frac{-4}{4} =-1 \ x_2= frac{-(-3)+ sqrt{49} }{2*2} = frac{3+7}{4} = frac{10}{4} =2.5 \ x eq -1 \ x=2.5[/latex] [latex] frac{3x+1}{x-1} = frac{2-3x}{x+2} \ x-1 eq 0~~~~x eq 1 \ x+3 eq 0~~~~x eq -3 \ (3x+1)(x+2)=(x-1)(2-3x) \ 3 x^{2} +x+6x+2=2x-2-3 x^{2} +3x \ 3 x^{2} +3 x^{2} +x+6x-2x-3x+2+2=0 \ 6 x^{2} +2x+4=0 \ 3 x^{2} +x+2=0 \ з=1^2-4*2*3=1-24=-23 extless 0[/latex] [latex] frac{3+x}{5x-4} = frac{2}{3} \ 3(3+x)=2(5x-4) \ 9+3x=10x-8 \ 3x-10x=-8-9 \ -7x=-17 \ x= frac{-17}{-7} \ x= frac{14+3}{7} \ x=2 frac{3}{7} [/latex]
