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

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 \( x = -5 \).