To solve the inequality \(x - 5 < \frac{x}{6}\), we want to isolate \(x\). Among the options provided, we need to identify the first appropriate step that simplifies the inequality correctly.
Let's analyze each option:
A. Divide both sides of the inequality by 6 to simplify: \(\frac{x}{6} - \frac{5}{6} < 6\)
This option is incorrect because it does not maintain the structure of the original inequality.
B. Divide both sides of the inequality by \(x\) to simplify: \(1 - \frac{5}{x} < 6\)
This option is also invalid because dividing by \(x\) could change the inequality depending on whether \(x\) is positive or negative.
C. Multiply both sides of the inequality by \(x\) to simplify: \(x - 5 < 6\)
This option is also not valid because multiplying by \(x\) can change the inequality sign, depending on the sign of \(x\).
D. Multiply both sides of the inequality by 6 to simplify: \(6x - 30 < x\)
This is a plausible choice as we can multiply through by 6, which is positive, thus keeping the direction of the inequality.
However, rather than multiplying by 6, a direct first step that maintains the integrity of the inequality would be to subtract \( \frac{x}{6} \) from both sides without multiplying or dividing.
Thus, you could ideally rewrite the inequality first:
- Start with: \( x - 5 < \frac{x}{6} \)
- Subtract \( \frac{x}{6} \) from both sides to get: \( x - \frac{x}{6} < 5 \)
The simplest first step among the provided options is D, as it maintains the inequality correctly:
So, the correct answer is D: Multiply both sides of the inequality by 6 to simplify: \(6x - 30 < x\).
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