1 Given: \( x \in\{-34 ;-8 ; 0 ; 7 ; 9\} \) hoose the correct value(s) of \( x \) from the list above so that \( \sqrt{\frac{36}{9+x}} \) is 4.1.1 \( \quad \) Rational 4.1.2 \( \quad \) Irrational 4.1.3 \( \quad \) Non - real 4.1.4 \( \quad \) Integer

To find the values of \( x \) that fit each condition, we can evaluate the expression \( \sqrt{\frac{36}{9+x}} \) based on the provided values of \( x \). First additional information: To determine rational or irrational outputs of the expression, remember that a square root results in a rational number if the radicand (the value under the square root) yields a perfect square. Thus, for \( 9+x \) in the denominator, the resulting fraction must simplify to a square of a rational number. For example, \( x = 0 \) gives us \( 9+0 = 9\), leading to \( \sqrt{4} = 2 \), which is rational. Second additional information: For non-real numbers, we must find \( x \) values that make \( 9+x \) negative, resulting in a negative denominator. Among the provided values, \( x = -34 \) gives \( 9 - 34 = -25\), making the expression non-real because you cannot take the square root of a negative number in the realm of real numbers! So, for non-real outcomes, \( x = -34 \) is a fitting choice.