[tex]a) \\\\2x*3x\neq0\\6x^2\neq0\\x^2\neq0\\x\neq0\\D\in R/\{0\}\\\\\\\frac{2x-1}{2x}-\frac{4x+3}{3x}=\frac{3x(2x-1)}{6x^2}-\frac{2x(4x+3)}{6x^2}=\frac{6x^2-3x-(8x^2+6x)}{6x^2}=\frac{6x^2-3x-8x^2-6x}{6x^2}=\frac{-2x^2-9x}{6x^2}=\frac{-(2x^2+9x)}{6x^2}=-\frac{2x^2+9x}{6x^2}=-\frac{x(2x+9)}{6x^2}=-\frac{2x+9}{6x}[/tex]
[tex]b)\\(x^2+2x)(2x-6)\neq0\\x^2+2x\neq0\\\Delta=2^2-4*1*0\\\sqrt{\Delta}=2\\x_1=\frac{-2-2}{2}=\frac{-4}2=-2\\x_2=\frac{-2+2}2=0\\\\\\2x-6\neq0 /+6\\2x \neq 6 /:2\\x \neq 3\\\\D\in R / \{-2, 0, 3\}[/tex]
[tex]\frac{x^2-9}{x^2+2x}*\frac{3x+6}{2x-6}=\frac{(x^2-9)(3x+6)}{(x^2+2x)(2x-6)}=\frac{3x^3+6x^2-27x-54}{2x^3-6x^2+4x^2-12x}=\frac{3x^3+6x^2-27x-54}{2x^3-2x^2-12x}=\frac{3(x^3+2x^2-9x-18)}{2x(x^2-x-6)}=\frac{3(x^2(x+2)-9(x+2))}{2x(x-3)(x+2)}=\frac{3(x^2-9)(x+2)}{2x(x-3)(x+2)}=\frac{3(x-3)(x+3)}{2x(x-3)}=\frac{3(x+3)}{2x}=\frac{3x+9}{2x}[/tex]
[tex]c)\\\\(x-2)(x+3)\neq0\\x-2 \neq0 /+2\\x \neq 2\\\\x+3 \neq 0 /-3\\x \neq -3\\\\D\in R / \{-3, 2\}[/tex]
[tex]\frac{3}{x-2}+\frac{4}{x+3}=\frac{3(x+3)}{(x-2)(x+3)}+\frac{4(x-2)}{(x-2)(x+3)}=\frac{3x+9+4x-8}{(x-2)(x+3)}=\frac{7x+1}{(x-2)(x+3)}[/tex]
[tex]d) \\\\(x^2-4)(x^2-3x)\neq0\\(x-2)(x+2)(x^2-3x) \neq 0\\x-2 \neq 0\\x \neq 2\\\\x+2 \neq 0\\x \neq -2\\\\x^2-3x \neq 0\\x_1=\frac{3-3}2=0\\x2=\frac{3+3}2=\frac62=3\\\\D\in R / \{-2, 0, 2, 3\}\\[/tex]
[tex]\frac{2x-6}{x^2-4}*\frac{3x+6}{x^2-3x}=\frac{2(x-3)*3(x+2)}{x(x-3)(x-2)(x+2)}=\frac{6(x-3)(x+2)}{x(x-3)(x-2)(x+2)}=\frac{6}{x(x-2)}[/tex]
[tex]e) \\(x-1)(x-5) \neq 0\\x-1 \neq 0 /+1\\x \neq 1\\\\x-5 \neq 0 /+5\\x \neq 5 \\\\D\in R / \{1 ,5\}[/tex]
[tex]\frac2{x+1}+\frac3{x-5}=\frac{2(x-5)}{(x+1)(x-5)}+\frac{3(x+1)}{(x+1)(x-5)}=\frac{2x-10+3x+3}{(x+1)(x-5)}=\frac{5x-7}{(x+1)(x-5)}[/tex]
[tex]f) \\\\3x*2x\neq 0\\6x^2 \neq 0 /:6\\x^2 \neq 0\\x \neq 0\\\\D\in R / \{0\}[/tex]
[tex]\frac{2x+1}{3x}-\frac{3x-2}{2x}=\frac{2x(2x+1)}{6x^2}-\frac{3x(3x-2)}{6x^2}=\frac{4x^2+2x-(9x^2-6x)}{6x^2}=\frac{4x^2+2x-9x^2+6x}{6x^2}=\frac{-5x^2+8x}{6x^2}=\frac{x(-5x+8)}{6x^2}=\frac{-5x+8}{6x}=-\frac{5x-8}{6x}[/tex]
