a)
[tex]x - 1 \geq 0\\x \geq 1 \\D: x \in \langle 1,\ \infty ) \\\\x \cdot \sqrt{x - 2} = 0 \iff x = 0 \lor \sqrt{x - 2} = 0\\x_0 = 2[/tex]
b)
[tex]7x^2 \neq 0\\\Delta = 0^2 - 4 \cdot 7 \cdot 0 = 0\\\sqrt{\Delta} = 0\\x = \frac{0}{4 \cdot 7} = 0\\D: x \in \mathbb{R} \setminus \{0\}\\\\\frac{(2x - 3)(x+5)}{7x^2} = 0 \iff 2x - 3 = 0 \lor x + 5 = 0\\x_0 \in \{-5,\ 1\frac{1}{2}\}[/tex]
c)
[tex]4 - x \geq 0\\4 \geq x\\D: x \in (-\infty,\ 4 \rangle \\\\(x^2 - 6x + 9) \cdot \sqrt{4 - x} = 0 \iff x^2 - 6x + 9 = 0 \lor \sqrt{4 - x} = 0\\\Delta = (-6)^2 - 4 \cdot 1 \cdot 9 = 36 - 36 = 0\\x = \frac{6}{4} = 1\frac{1}{2} \\x_0 \in \{1\frac{1}{2} ,\ 4\}[/tex]
d)
[tex]3x - 9 \neq 0\\D: x\in \mathbb{R} \setminus \{3\}\\\\\frac{x^2+1}{3x-9} = 0 \iff x^2 + 1 = 0\\\Delta > 0 \implies x \in \varnothing \\\\x_0 \in \varnothing[/tex]
