1. **State the problem:** Find the set $A=\{x \in \mathbb{Z} \mid (x-1)(x-2)(x+\frac{1}{2})=0\}$. This means we want all integer values of $x$ that satisfy the equation. 2. **Use the zero product property:** If a product of factors equals zero, then at least one of the factors must be zero. So, $$ (x-1)(x-2)(x+\frac{1}{2})=0 \implies x-1=0 \text{ or } x-2=0 \text{ or } x+\frac{1}{2}=0 $$ 3. **Solve each factor:** - $x-1=0 \implies x=1$ - $x-2=0 \implies x=2$ - $x+\frac{1}{2}=0 \implies x=-\frac{1}{2}$ 4. **Check integer condition:** Since $x \in \mathbb{Z}$ (integers), $x=-\frac{1}{2}$ is not an integer and must be excluded. 5. **Final solution:** $$ A = \{1, 2\} $$ These are the integer values of $x$ that satisfy the equation.