d)
\frac{-3x+4}{x-1}=\frac{4}{x-4}-3
Dziedzina
x-1\ne0=>x\ne 1
x-4\ne 0=>x\ne 4
D: x\in \mathbb R \backslash \{1, 4\}
\frac{-3x+4}{x-1}=\frac{4(x-4)}{x-4}
\frac{-3x+4}{x-1}=\frac{4-3x+12}{x-4}
\frac{-3x+4}{x-1}=\frac{16-3x}{x-4} Korzystam z własności proporcji - mnożę “na krzyż”
(-3x+4)(x-4)=(x-1)(16-3x)
-3x^2+12x+4x-16=16x-3x^2-16+3x
-3x^2+16x-16-16x+3x^2+16-3x=0
-3x=0 \ |:(-3)
x=0
e)
\frac{x+1}{2x+3}=\frac{2x+1}{3x+2}
Dziedzina
2x+3\ne 0=> 2x\ne -3=> x\ne -\frac{3}{2}
3x+2\ne 0=> 3x\ne -2=> x\ne -\frac{2}{3}
D: x\in \mathbb R \backslash \{-\frac{3}{2}, \ -\frac{2}{3}\}
(2x+1)(2x+3)=(x+1)(3x+2)
4x^2+6x+2x+3=3x^2+2x+3x+2
4x^2+8x+3-3x^2-2x-3x-2=0
x^2+3x+1=0
a=1, b=3, c=-1
\Delta=b^2-4ac=9-4\cdot 1 \cdot 1=5
\sqrt\Delta=\sqrt5
x_1=\frac{-b-\sqrt\Delta}{2a}=\frac{-3-5}{2\cdot 1 }=\frac{-3-\sqrt5}{2}
x_2=\frac{-b+\sqrt\Delta}{2a}=\frac{-3-\sqrt5}{2\cdot 1}=\frac{\sqrt5-3}{2}
f)
\frac{x+5}{2x-6}=\frac{2x-7}{3x-9}
\frac{x+5}{2(x-3)}=\frac{2x-7}{3(x-3)} \ |*(x-3)
Dziedzina:
2x-3\ne 0 => x-3\ne0 => x\ne 3
3(x-3)\ne 0 => x\ne 3
D: x\in \mathbb R \backslash \{3\}
\frac{x+5}{2}=\frac{2x-7}{3} \ |*6
3(x+5)=2(2x-7)
3x+15=4x-14
-x=-29 \ |*(-1)
x=29