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

Question

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

UpStudy AI · Accepted Answer

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         |
For each value of $$ x $$ in $$\{-14, -11, -5, 0, 5, 11, 14\}$$, the expression $$ A = \sqrt{\frac{9}{11 - x}} $$ results in: - **Rational**: $$ x = -14, -5 $$ - **Irrational**: $$ x = -11, 0, 5 $$ - **Undefined**: $$ x = 11 $$ - **Non-Real**: $$ x = 14 $$

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