Question

Use algebraic​ and/or graphical methods to solve the inequality.
9 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 \( 9x^2 - x^4 \geq 0 \), we can first rewrite it in a more manageable form.

1. Rewrite the inequality:
\[
-x^4 + 9x^2 \geq 0
\]
This can also be written as:
\[
x^4 - 9x^2 \leq 0
\]

2. Factor the left-hand side:
\[
x^2(x^2 - 9) \leq 0
\]
This further factors to:
\[
x^2(x - 3)(x + 3) \leq 0
\]

3. Find the critical points:
The critical points occur where the expression is equal to zero:
\[
x^2 = 0 \implies x = 0
\]
\[
x - 3 = 0 \implies x = 3
\]
\[
x + 3 = 0 \implies x = -3
\]
The critical points are \( x = -3, 0, 3 \).

4. Determine the signs in the intervals:
We will test the sign of \( x^2(x - 3)(x + 3) \) in the intervals determined by the critical points: \( (-\infty, -3) \), \( (-3, 0) \), \( (0, 3) \), and \( (3, \infty) \).

- For \( x < -3 \) (e.g., \( x = -4 \)):
\[
(-4)^2(-4 - 3)(-4 + 3) = 16(-7)(-1) > 0
\]
- For \( -3 < x < 0 \) (e.g., \( x = -1 \)):
\[
(-1)^2(-1 - 3)(-1 + 3) = 1(-4)(2) < 0
\]
- For \( 0 < x < 3 \) (e.g., \( x = 1 \)):
\[
(1)^2(1 - 3)(1 + 3) = 1(-2)(4) < 0
\]
- For \( x > 3 \) (e.g., \( x = 4 \)):
\[
(4)^2(4 - 3)(4 + 3) = 16(1)(7) > 0
\]

5. Identify intervals where the product is less than or equal to zero:
The product \( x^2(x - 3)(x + 3) \leq 0 \) is satisfied in the intervals where it is negative or zero:
- The expression is zero at \( x = -3, 0, 3 \).
- It's negative in the intervals \((-3, 0)\) and \((0, 3)\).

6. Combine the intervals:
Including the endpoints where the expression is equal to zero, the solution in interval notation is:
\[
[-3, 0] \cup [0, 3]
\]

Thus, depending on the structure of the choices provided, if we write them down, the solution is:

**Choice B**:
The solutions are the values of \( x \) in the interval:
\[
[-3, 0] \cup [0, 3]
\]