To solve the inequality \( x - 5 < \frac{x}{6} \), we want to isolate \( x \) and get the terms involving \( x \) on one side.
Let's analyze each option to determine which is the most suitable first step:
A. Divide both sides of the inequality by 6 to simplify: \( \frac{x}{6} - \frac{5}{6} < 6 \)
This is not a correct approach, because dividing by 6 will make the inequality complicated and doesn't lead directly to a clear form.
B. Divide both sides of the inequality by \( x \) to simplify: \( 1 - \frac{5}{x} < 6 \)
This is also not a good approach, as it assumes \( x \neq 0 \) and introduces complications without directly solving for \( x \).
C. Multiply both sides of the inequality by \( x \) to simplify: \( x - 5 < 6 \)
This could be problematic unless we know the signs of \( x \) since multiplying or dividing by a negative number reverses the inequality.
D. Multiply both sides of the inequality by 6 to simplify: \( 6x - 30 < x \)
This is a valid approach, and it will preserve the inequality since we are multiplying by a positive number.
Given these analyses, option D is the first and correct step to simplify the inequality without introducing additional complications related to the sign of \( x \).
Thus, the answer is: D. Multiply both sides of the inequality by 6 to simplify: \( 6x - 30 < x \).