To solve the inequality \( x - 6 < x^4 \), we want to get all the terms on one side. A good first step would be to rewrite it as:
\[ x^4 - x + 6 > 0 \]
However, among the responses provided, a statement that shows a good first step is:
"Multiply both sides by 4 to clear the inequality of fractions: 4x - 24 < x."
This response is less appropriate, given the original inequality doesn’t have fractions to clear, so let's assess the alternatives.
Regarding the correct solution for \( x - 6 < x^4 \):
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Rearrange the inequality: \[ x^4 - x + 6 > 0 \]
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Analyze the inequality \( x^4 - x + 6 > 0 \):
- \( x^4 \) is always non-negative for real \( x \).
- The expression \( -x + 6 \) shifts downward (negative contribution when \( x > 6 \)).
- The fourth-degree polynomial will dominate and remain positive as \( x^4 \) grows faster than the linear term \( -x + 6 \).
Thus, after analyzing the polynomial behavior, we find that the inequality holds for all real numbers.
Correct conclusion:
\[ x^4 - x + 6 > 0 \text{ for all real } x. \]
However, reviewing the original choices carefully indicates that they may not directly lead to a correct simplification step. The simplest approach might actually involve testing values or roots rather than algebraic transformations in many cases.
In this situation, the systematic constant growth of \( x^4 \) implies:
- The inequality \( x - 6 < x^4 \) holds true for all \( x \).
Thus, no specific conditions are solved for \( x \) in the positive side – every real number \( x \) will hold true.