Question

Find and correct the error.


x4+6>3

x+6>12

x>6

(1 point)
Responses

With inequalities you always flip the symbol at the end - the true answer should be x < 6
With inequalities you always flip the symbol at the end - the true answer should be x < 6

6 should have been subtracted first before multiplying by 4 - the true answer should be x > -3/4
6 should have been subtracted first before multiplying by 4 - the true answer should be x > -3/4

6 should have been added first before multiplying by 4 - the true answer should be x > 36
6 should have been added first before multiplying by 4 - the true answer should be x > 36

6 should have been subtracted first before multiplying by 4 - the true answer should be x > -12
6 should have been subtracted first before multiplying by 4 - the true answer should be x > -12

Answer 1

Let's analyze the original inequality and the steps taken to solve it.

The original inequality given is:

1. \( x^4 + 6 > 3 \)

To solve for \( x \):

1. Subtract 6 from both sides:
\[
x^4 > 3 - 6
\]
This simplifies to:
\[
x^4 > -3
\]

2. 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:

2. \( x + 6 > 12 \)

To solve for \( x \):

1. 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.