Question

Solving the inequality for x, which statement shows a good first step and which is the correct solution?

x−6<x4
(2 points)
Responses

Divide both sides of the inequality by x to simplify: 1−6x<4
x<6


Divide both sides of the inequality by x to simplify: 1−6x<4
x<6

Multiply both sides by 4 to clear the inequality of fractions: 4x−24<x


Multiply both sides by 4 to clear the inequality of fractions: 4x−24<x

Multiply both sides by x to get the "x" terms on the same side of the inequality: x−6<4




Multiply both sides by x to get the "x" terms on the same side of the inequality: x−6<4


Divide both sides of the inequality by 4 to simplify: x4−32<x

Answer 1

To solve the inequality \(x - 6 < x^4\), a good first step would be to rearrange the inequality so that all terms are on one side.

1. First, let's move everything to one side of the inequality:
\[
x - 6 - x^4 < 0
\]
This can be rewritten as:
\[
-x^4 + x - 6 < 0
\]
Or equivalently:
\[
x^4 - x + 6 > 0
\]
This form makes it easier to analyze the roots and the behavior of the polynomial.

Given the choices in your question, none of them are the best first steps to rearranging the inequality directly. However, involving multiplying the inequality by a term (like x) can also pose issues depending on the value of x (as it could flip the inequality sign if x is negative).

But to answer your question directly regarding a good first step, multiplying by x and attempting to manipulate it further is one approach, though it must be handled carefully. The best step generally would be to rearrange the equation as shown above.

As for the correct solution, analyzing the function \(x^4 - x + 6\):

1. Since \(x^4\) dominates and is always non-negative for all real x, we notice that the whole function is always greater than zero. Thus, the inequality \(x^4 - x + 6 > 0\) holds true for all real x.

Therefore, the conclusion is:
- The inequality \(x - 6 < x^4\) is true for all x in the real numbers.

If we summarize:
- The best first step is to rewrite \(x - 6 < x^4\) in a standard polynomial form.
- The solution shows that the inequality holds for all x.