To solve the inequality \(x - 5 < \frac{x}{6}\), it's important to isolate \(x\) on one side of the inequality. Let's analyze each option:
A. Divide both sides of the inequality by 6 to simplify: \( \frac{x}{6} - \frac{5}{6} < 6 \)
This step is not appropriate because it does not correctly simplify the original inequality.
B. Divide both sides of the inequality by \(x\) to simplify: \(1 - \frac{5}{x} < 6\)
This is not a safe step unless \(x\) is guaranteed to be positive, as dividing by a variable can change the direction of the inequality.
C. Multiply both sides of the inequality by \(x\) to simplify: \(x - 5 < 6\)
This is incorrect for the same reason as option B; multiplying by \(x\) can be risky unless \(x\) is known to be positive.
D. Multiply both sides of the inequality by 6 to simplify: \(6x - 30 < x\)
This is a valid step and maintains the direction of the inequality because you're multiplying by a positive constant.
Given these evaluations, the best first step to solve the inequality \(x - 5 < \frac{x}{6}\) is D: Multiply both sides of the inequality by 6 to simplify it to \(6x - 30 < x\).