7. For which values of \( x ; x \in\{-11 ;-5 ; 0 ; 15\} \) is \( \sqrt{\frac{9}{11-x}} \) \( \begin{array}{lll}\text { a) a rational number? } & \text { b) an irrational number? } & \text { c) not defined? }\end{array} \)

To determine for which values of \( x \) the expression \( \sqrt{\frac{9}{11-x}} \) is a rational number, irrational number, or not defined, let's analyze the expression for each given \( x \): 1. **For \( x = -11 \)**: \[ \sqrt{\frac{9}{11 - (-11)}} = \sqrt{\frac{9}{22}} = \frac{3}{\sqrt{22}} \text{ (irrational)} \] 2. **For \( x = -5 \)**: \[ \sqrt{\frac{9}{11 - (-5)}} = \sqrt{\frac{9}{16}} = \frac{3}{4} \text{ (rational)} \] 3. **For \( x = 0 \)**: \[ \sqrt{\frac{9}{11 - 0}} = \sqrt{\frac{9}{11}} \text{ (irrational)} \] 4. **For \( x = 15 \)**: \[ \sqrt{\frac{9}{11 - 15}} = \sqrt{\frac{9}{-4}} \text{ (not defined, as we cannot take the square root of a negative number)} \] **Summary**: - Rational number: \( x = -5 \) - Irrational numbers: \( x = -11 \) and \( x = 0 \) - Not defined: \( x = 15 \)