E) \( \frac{4 x-1}{x+5}=0 \)

To find the values of \( x \) in the equation \( \frac{4x-1}{x+5}=0 \), we focus on the numerator since a fraction equals zero when its numerator is zero. Setting \( 4x - 1 = 0 \) leads us to \( 4x = 1 \), and thus \( x = \frac{1}{4} \). To ensure this answer is valid, we must check if the denominator \( x + 5 \) is non-zero at \( x = \frac{1}{4} \), and indeed it is (since \( \frac{1}{4} + 5 = \frac{21}{4} \)). So, \( x = \frac{1}{4} \) is our solution! Now, remember that in any equation involving fractions, it’s crucial to consider the restrictions presented by the denominator to avoid undefined values. Keep in mind that if \( x + 5 \) had equaled zero, the solution wouldn’t have been valid. It’s a great reminder to always check both parts of a rational equation!