Question

Use algebraic​ and/or graphical methods to solve the inequality.
10 x cubed minus x Superscript 4less than0
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The solutions are the values of x in the interval
  
enter your response here.
​(Type your answer in interval​ notation.)
B.
The​ solution(s) is/are xequals
  
enter your response here.
​(Use a comma to separate answers as​ needed.)
C.
There is no solution.

Answer 1

To solve the inequality \( 10x^3 - x^4 < 0 \), we can start by factoring the left-hand side.

First, we can factor out \( x^3 \):

\[
10x^3 - x^4 = x^3(10 - x) < 0
\]

Next, we set the factored expression equal to zero to find the critical points:

\[
x^3(10 - x) = 0
\]

This gives us two parts to consider:

1. \( x^3 = 0 \) which yields \( x = 0 \)
2. \( 10 - x = 0 \) which yields \( x = 10 \)

Now, we have two critical points: \( x = 0 \) and \( x = 10 \). These critical points divide the number line into three intervals:

1. \( (-\infty, 0) \)
2. \( (0, 10) \)
3. \( (10, \infty) \)

Next, we will test a point from each interval in the inequality \( x^3(10 - x) < 0 \) to see where the expression is negative.

1. **Interval \( (-\infty, 0) \)**: Choose \( x = -1 \)
\[
(-1)^3(10 - (-1)) = -1(11) = -11 < 0 \quad (\text{True})
\]

2. **Interval \( (0, 10) \)**: Choose \( x = 5 \)
\[
(5)^3(10 - 5) = 125(5) = 625 > 0 \quad (\text{False})
\]

3. **Interval \( (10, \infty) \)**: Choose \( x = 11 \)
\[
(11)^3(10 - 11) = 1331(-1) = -1331 < 0 \quad (\text{True})
\]

From testing these intervals, we see that the expression \( x^3(10 - x) < 0 \) is satisfied in the intervals \( (-\infty, 0) \) and \( (10, \infty) \).

Thus, the solution in interval notation is:

\[
(-\infty, 0) \cup (10, \infty)
\]

**Part 1 Answer:**

A. The solutions are the values of \( x \) in the interval
\[
(-\infty, 0) \cup (10, \infty)
\]

So the answer is:
\[
(-\infty, 0) \cup (10, \infty)
\]