3. If $$ R=\frac{\sqrt{x+5}}{x+3} $$, for which values of $$ x $$ will $$ R $$ be: a) real b) non real c) undefined d) equal to zero

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To analyze the expression $$ R = \frac{\sqrt{x+5}}{x+3} $$, we need to consider the conditions under which $$ R $$ is real, non-real, undefined, or equal to zero.

### a) For $$ R $$ to be real:
1. The square root $$ \sqrt{x+5} $$ must be defined, which requires:
   $$
   x + 5 \geq 0 \implies x \geq -5
   $$
2. The denominator $$ x + 3 $$ must not be zero, which requires:
   $$
   x + 3 \neq 0 \implies x \neq -3
   $$

Combining these conditions, $$ R $$ is real for:
$$
x \geq -5 \quad \text{and} \quad x \neq -3
$$

### b) For $$ R $$ to be non-real:
The expression $$ R $$ will be non-real if the square root is not defined, which occurs when:
$$
x + 5 < 0 \implies x < -5
$$

### c) For $$ R $$ to be undefined:
The expression $$ R $$ is undefined when the denominator is zero:
$$
x + 3 = 0 \implies x = -3
$$

### d) For $$ R $$ to be equal to zero:
The expression $$ R $$ is equal to zero when the numerator is zero:
$$
\sqrt{x+5} = 0 \implies x + 5 = 0 \implies x = -5
$$

### Summary of Results:
- a) $$ R $$ is real for $$ x \geq -5 $$ and $$ x \neq -3 $$.
- b) $$ R $$ is non-real for $$ x < -5 $$.
- c) $$ R $$ is undefined for $$ x = -3 $$.
- d) $$ R $$ is equal to zero for $$ x = -5 $$.
- $$ R $$ is real when $$ x \geq -5 $$ and $$ x \neq -3 $$. - $$ R $$ is non-real when $$ x < -5 $$. - $$ R $$ is undefined when $$ x = -3 $$. - $$ R $$ equals zero when $$ x = -5 $$.

If , for which values of will be:
a) real
b) non real
c) undefined
d) equal to zero

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If , for which values of will be: a) real b) non real c) undefined d) equal to zero