Find the domain of the expression. $$ \begin{array}{l}\frac{\sqrt{x-4}}{x+6} \ x \neq 4 \ x-6 \ x \neq-6 \ \text { all real numbers } \ x \geq 4\end{array} $$

Question

Find the domain of the expression. $$ \begin{array}{l}\frac{\sqrt{x-4}}{x+6} \ x \neq 4 \ x>-6 \ x \neq-6 \ \text { all real numbers } \ x \geq 4\end{array} $$

UpStudy AI · Accepted Answer

To determine the domain of the expression:

$$
\frac{\sqrt{x - 4}}{x + 6}
$$

we need to consider the restrictions imposed by both the square root in the numerator and the denominator.

1. **Square Root Restriction:**
   - The expression inside the square root must be non-negative.
   $$
   x - 4 \geq 0 \implies x \geq 4
   $$

2. **Denominator Restriction:**
   - The denominator cannot be zero.
   $$
   x + 6 \neq 0 \implies x \neq -6
   $$
   However, since $$ x \geq 4 $$, the condition $$ x \neq -6 $$ is automatically satisfied because $$ -6 < 4 $$.

Therefore, the only restriction that affects the domain is $$ x \geq 4 $$.

**Domain:** $$ x \geq 4 $$

**Answer:** $$ x \geq 4 $$
The domain of the expression is all real numbers $$ x $$ such that $$ x \geq 4 $$.

Find the domain of the expression.

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