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

Question

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:
\[ [-3, 0] \cup [0, 3] \]