\( \begin{array}{ll}\text { 1. Consider : } A=\sqrt{\frac{9}{11-x}} \text { if } \\ & x \in(-14 ;-11 ;-5 ; 0 ; 5 ; 11 ; 14) ; \text { which value(s) of } \mathrm{x} \text { will make } \mathrm{A}: \\ \text { 1.1. } & \text { Rational } \\ \text { 1.2. } & \text { Irrational } \\ \text { 1.3. } & \text { Undefined } \\ \text { 1.4. } & \text { Non-Real }\end{array} \)

Let's analyze the function \( A = \sqrt{\frac{9}{11 - x}} \) for each value of \( x \) in the set \(\{-14, -11, -5, 0, 5, 11, 14\}\). ### Step 1: Evaluate the Expression for Each \( x \) 1. **For \( x = -14 \):** \[ A = \sqrt{\frac{9}{11 - (-14)}} = \sqrt{\frac{9}{25}} = \frac{3}{5} \quad \text{(Rational)} \] 2. **For \( x = -11 \):** \[ A = \sqrt{\frac{9}{11 - (-11)}} = \sqrt{\frac{9}{22}} = \frac{3}{\sqrt{22}} \quad \text{(Irrational)} \] 3. **For \( x = -5 \):** \[ A = \sqrt{\frac{9}{11 - (-5)}} = \sqrt{\frac{9}{16}} = \frac{3}{4} \quad \text{(Rational)} \] 4. **For \( x = 0 \):** \[ A = \sqrt{\frac{9}{11 - 0}} = \sqrt{\frac{9}{11}} = \frac{3}{\sqrt{11}} \quad \text{(Irrational)} \] 5. **For \( x = 5 \):** \[ A = \sqrt{\frac{9}{11 - 5}} = \sqrt{\frac{9}{6}} = \sqrt{\frac{3}{2}} = \frac{\sqrt{6}}{2} \quad \text{(Irrational)} \] 6. **For \( x = 11 \):** \[ A = \sqrt{\frac{9}{11 - 11}} = \sqrt{\frac{9}{0}} \quad \text{(Undefined)} \] 7. **For \( x = 14 \):** \[ A = \sqrt{\frac{9}{11 - 14}} = \sqrt{\frac{9}{-3}} = \sqrt{-3} \quad \text{(Non-Real)} \] ### Step 2: Summary of Results 1. **1.1. Values of \( x \) that make \( A \) **Rational**:** - \( x = -14 \) - \( x = -5 \) 2. **1.2. Values of \( x \) that make \( A \) **Irrational**:** - \( x = -11 \) - \( x = 0 \) - \( x = 5 \) 3. **1.3. Values of \( x \) that make \( A \) **Undefined**:** - \( x = 11 \) 4. **1.4. Values of \( x \) that make \( A \) **Non-Real**:** - \( x = 14 \) ### Final Classification | \( x \) | \( A = \sqrt{\frac{9}{11 - x}} \) | Classification | |----------|------------------------------------|------------------| | \(-14\) | \( \frac{3}{5} \) | Rational | | \(-11\) | \( \frac{3}{\sqrt{22}} \) | Irrational | | \(-5\) | \( \frac{3}{4} \) | Rational | | \(0\) | \( \frac{3}{\sqrt{11}} \) | Irrational | | \(5\) | \( \frac{\sqrt{6}}{2} \) | Irrational | | \(11\) | Undefined | Undefined | | \(14\) | \( \sqrt{-3} \) | Non-Real |