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

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

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Let's evaluate the expression $$ \sqrt{\frac{9}{11 - x}} $$ for each value of $$ x \in \{-11, -5, 0, 15\} $$. ### Step 1: Determine When the Expression is Defined The expression $$ \sqrt{\frac{9}{11 - x}} $$ is defined when the denominator $$ 11 - x $$ is **not zero** and the entire fraction under the square root is **non-negative**. 1. **Denominator not zero**: $$ 11 - x \neq 0 \implies x \neq 11 $$ None of the provided values ($$-11, -5, 0, 15$$) equal 11, so this condition is satisfied for all. 2. **Fraction is non-negative**: $$ \frac{9}{11 - x} \geq 0 $$ Since 9 is positive, the sign of the fraction depends on $$ 11 - x $$: - $$ 11 - x > 0 \implies x < 11 $$ - $$ 11 - x < 0 \implies x > 11 $$ Therefore: - For $$ x = -11, -5, 0 $$: $$ x < 11 $$, so the fraction is positive. - For $$ x = 15 $$: $$ x > 11 $$, so the fraction is negative, making the square root undefined in the real numbers. ### Step 2: Evaluate the Expression for Each $$ x $$ 1. **$$ x = -11 $$**: $$ \sqrt{\frac{9}{11 - (-11)}} = \sqrt{\frac{9}{22}} = \frac{3}{\sqrt{22}} \approx 0.64 \text{ (Irrational)} $$ 2. **$$ x = -5 $$**: $$ \sqrt{\frac{9}{11 - (-5)}} = \sqrt{\frac{9}{16}} = \frac{3}{4} = 0.75 \text{ (Rational)} $$ 3. **$$ x = 0 $$**: $$ \sqrt{\frac{9}{11 - 0}} = \sqrt{\frac{9}{11}} = \frac{3}{\sqrt{11}} \approx 0.90 \text{ (Irrational)} $$ 4. **$$ x = 15 $$**: $$ \sqrt{\frac{9}{11 - 15}} = \sqrt{\frac{9}{-4}} \text{ is not defined in the real numbers} $$ ### Summary **a) Rational Number:** - $$ x = -5 $$ **b) Irrational Number:** - $$ x = -11 $$ - $$ x = 0 $$ **c) Not Defined:** - $$ x = 15 $$ - **a) Rational Number:** $$ x = -5 $$ - **b) Irrational Number:** $$ x = -11 $$ and $$ x = 0 $$ - **c) Not Defined:** $$ x = 15 $$

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