For which value(s) of \( x \) will \( \frac{2 x-4}{\sqrt{3 x-1}} \) 1.2 .1 be imaginary? 1.2 .2 be undefined?

To determine the values of \( x \) for which the expression \( \frac{2x-4}{\sqrt{3x-1}} \) is imaginary or undefined, we need to analyze the conditions for both cases. ### 1.2.1 When will the expression be imaginary? The expression will be imaginary if the denominator is zero, as the square root of a negative number is imaginary. Therefore, we need to find when: \[ \sqrt{3x - 1} < 0 \] However, the square root function is defined only for non-negative values. Thus, \( \sqrt{3x - 1} \) can never be negative. Therefore, the expression cannot be imaginary for any real value of \( x \). ### 1.2.2 When will the expression be undefined? The expression will be undefined when the denominator is zero. This occurs when: \[ \sqrt{3x - 1} = 0 \] Squaring both sides gives: \[ 3x - 1 = 0 \] Solving for \( x \): \[ 3x = 1 \\ x = \frac{1}{3} \] Thus, the expression \( \frac{2x-4}{\sqrt{3x-1}} \) is undefined at \( x = \frac{1}{3} \). ### Summary: - The expression is never imaginary for real values of \( x \). - The expression is undefined at \( x = \frac{1}{3} \).