Question

Use algebraic​ and/or graphical methods to solve the inequality.
121 x squared minus x Superscript 4 Baseline greater than or equals 0
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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​ solution(s) is/are xequals
  
enter your response here.
​(Use a comma to separate answers as​ needed.)
B.
The solutions are the values of x in the interval
  
enter your response here.
​(Type your answer in interval​ notation.)
C.
There is no solution.

Answer 1

To solve the inequality \( 121x^2 - x^4 \geq 0 \), let's first rewrite the expression:

\[
-x^4 + 121x^2 \geq 0
\]

To make it easier to factor, let's rearrange it:

\[
121x^2 - x^4 \geq 0
\]

Next, we can factor out \( x^2 \):

\[
x^2(121 - x^2) \geq 0
\]

Now, we can set each factor equal to zero to find the critical points:

1. \( x^2 = 0 \) gives \( x = 0 \).
2. \( 121 - x^2 = 0 \) gives \( x^2 = 121 \), hence \( x = \pm 11 \).

So the critical points are \( x = -11, 0, 11 \).

Next, we will use these points to test the intervals around them to determine where the product is non-negative. The intervals are:

1. \( (-\infty, -11) \)
2. \( (-11, 0) \)
3. \( (0, 11) \)
4. \( (11, \infty) \)

**Testing the intervals:**

1. **For the interval \((- \infty, -11)\)**, choose \( x = -12 \):
\[
(-12)^2(121 - (-12)^2) = 144(121 - 144) = 144(-23) < 0
\]
This interval does not satisfy the inequality.

2. **For the interval \((-11, 0)\)**, choose \( x = -1 \):
\[
(-1)^2(121 - (-1)^2) = 1(121 - 1) = 1(120) > 0
\]
This interval satisfies the inequality.

3. **For the interval \( (0, 11)\)**, choose \( x = 1 \):
\[
(1)^2(121 - (1)^2) = 1(121 - 1) = 1(120) > 0
\]
This interval satisfies the inequality.

4. **For the interval \( (11, \infty)\)**, choose \( x = 12 \):
\[
(12)^2(121 - (12)^2) = 144(121 - 144) = 144(-23) < 0
\]
This interval does not satisfy the inequality.

**Endpoints:**
- At \( x = -11 \):
\[
(-11)^2(121 - (-11)^2) = 121(0) = 0
\]
This satisfies the inequality.

- At \( x = 0 \):
\[
(0)^2(121 - (0)^2) = 0(121) = 0
\]
This satisfies the inequality.

- At \( x = 11 \):
\[
(11)^2(121 - (11)^2) = 121(0) = 0
\]
This satisfies the inequality.

**Final solution:**

The intervals satisfying the inequality are:

\[
[-11, 0] \cup [0, 11]
\]

Since the point at \( x=0 \) is included in both intervals, we can simplify this to:

\[
[-11, 11]
\]

The answer is:

**Part B: The solutions are the values of \( x \) in the interval \([-11, 11]\).**