Let's analyze the original inequality and the steps taken to solve it.
The original inequality given is:
- \( x^4 + 6 > 3 \)
To solve for \( x \):
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Subtract 6 from both sides: \[ x^4 > 3 - 6 \] This simplifies to: \[ x^4 > -3 \]
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The fourth power of any real number \( x \) is always non-negative, thus \( x^4 \) will always be greater than -3 for all real numbers \( x \). Therefore: \[ x \in \mathbb{R} \quad (\text{all real numbers satisfy the inequality}) \]
Now the second provided inequality is:
- \( x + 6 > 12 \)
To solve for \( x \):
- Subtract 6 from both sides: \[ x > 12 - 6 \] This simplifies to: \[ x > 6 \]
The last step is correct, as well as the solution \( x > 6 \).
Upon analyzing the options provided:
- "With inequalities you always flip the symbol at the end - the true answer should be x < 6" (Incorrect)
- "6 should have been subtracted first before multiplying by 4 - the true answer should be x > -3/4" (Incorrect)
- "6 should have been added first before multiplying by 4 - the true answer should be x > 36" (Incorrect)
- "6 should have been subtracted first before multiplying by 4 - the true answer should be x > -12" (Incorrect)
None of the response options corrects an error that doesn't exist in the process for the second inequality; the solution \( x > 6 \) stands accurate.
So, the conclusion is:
- The derived inequality \( x > 6 \) is correct and there was actually no error in solving it.