Use algebraic​ and/or graphical methods to solve the inequality.

Question

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 \([-11, 11]\).