To solve the inequality \( x - 5 < \frac{x}{6} \), let's analyze the options:
A. Divide both sides of the inequality by 6 to simplify:
This would give us \( \frac{x-5}{6} < \frac{x}{36} \), which is more complicated than necessary.
B. Divide both sides of the inequality by x to simplify:
This is not valid when \( x = 0 \) or \( x < 0 \) because dividing by a variable can change the direction of the inequality based on the sign of \( x \).
C. Multiply both sides of the inequality by x to simplify:
Multiplying by \( x \) considers whether \( x \) is positive or negative and is not a valid step without determining the sign of \( x \).
D. Multiply both sides of the inequality by 6 to simplify:
This maintains the integrity of the inequality and does not change the direction because 6 is positive. Thus, it results in:
\[ 6(x - 5) < x \]
or
\[ 6x - 30 < x \],
This step will allow us to isolate \( x \) easily.
The correct first step is: D. Multiply both sides of the inequality by 6 to simplify: 6x - 30 < x.