1. **Problem statement:** Find the domain (conditions for which the function is defined) of each rational expression. 2. **Key rule:** A rational expression is defined when its denominator is not zero. 3. **Solve each part:** - a) $A(x) = \frac{5x - 6}{3x}$ - Denominator: $3x \neq 0 \Rightarrow x \neq 0$ - Domain: all real $x$ except $0$ - b) $B(x) = \frac{x}{6y}$ - Denominator: $6y \neq 0 \Rightarrow y \neq 0$ - Domain: all real $y$ except $0$ - c) $C(x) = \frac{5x - 1}{3(x + 1)}$ - Denominator: $3(x + 1) \neq 0 \Rightarrow x + 1 \neq 0 \Rightarrow x \neq -1$ - Domain: all real $x$ except $-1$ - d) $D(x) = \frac{8}{x^2 - 4}$ - Denominator: $x^2 - 4 \neq 0$ - Factor denominator: $x^2 - 4 = (x - 2)(x + 2)$ - So, $(x - 2)(x + 2) \neq 0 \Rightarrow x \neq 2$ and $x \neq -2$ - Domain: all real $x$ except $2$ and $-2$ **Final answers:** - a) $x \neq 0$ - b) $y \neq 0$ - c) $x \neq -1$ - d) $x \neq 2, -2$