Let \( f(x)=\frac{(x+6)}{(x-3)(2 x+5)} \) Find the domain in interval notation Question Help: Use -oo for \( -\infty \), oo for \( \infty \), U for union. Submit Question

To determine the domain of the function \( f(x) = \frac{x + 6}{(x - 3)(2x + 5)} \), we need to identify all real numbers \( x \) for which the denominator is not zero, since division by zero is undefined. 1. **Set the denominator not equal to zero:** \[ (x - 3)(2x + 5) \neq 0 \] 2. **Find the values that make the denominator zero:** \[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \] \[ 2x + 5 = 0 \quad \Rightarrow \quad x = -\frac{5}{2} \] 3. **Exclude these values from the real numbers:** \[ x \neq 3 \quad \text{and} \quad x \neq -\frac{5}{2} \] 4. **Express the domain in interval notation:** \[ (-\infty, -\frac{5}{2}) \cup (-\frac{5}{2}, 3) \cup (3, \infty) \] **Final Answer:** \[ (-\infty, -\frac{5}{2}) \cup (-\frac{5}{2},\, 3) \cup (3,\, \infty) \]