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

Question

49 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 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 \(49x^2 - x^4 \geq 0\), we can start by rearranging it:

\[ -x^4 + 49x^2 \geq 0 \]

This can be rewritten as:

\[ x^4 - 49x^2 \leq 0 \]

Now, we can factor the left side:

\[ x^2(x^2 - 49) \leq 0 \]

Next, we can further factor \(x^2 - 49\) using the difference of squares:

\[ x^2(x - 7)(x + 7) \leq 0 \]

Now we will find the critical points by setting each factor to zero:

1. \(x^2 = 0 \Rightarrow x = 0\)
2. \(x - 7 = 0 \Rightarrow x = 7\)
3. \(x + 7 = 0 \Rightarrow x = -7\)

Thus, the critical points are \(x = -7\), \(x = 0\), and \(x = 7\). These points divide the real number line into intervals:

1. \((- \infty, -7)\)
2. \((-7, 0)\)
3. \((0, 7)\)
4. \((7, \infty)\)

Next, we can test points from each interval to see where the expression \(x^2(x - 7)(x + 7)\) is less than or equal to zero.

1. Test a point in \((- \infty, -7)\), e.g. \(x = -8\): \[ (-8)^2(-8 - 7)(-8 + 7) = 64(-15)(-1) = 960 > 0 \]

2. Test a point in \((-7, 0)\), e.g. \(x = -1\): \[ (-1)^2(-1 - 7)(-1 + 7) = 1(-8)(6) = -48 < 0 \]

3. Test a point in \((0, 7)\), e.g. \(x = 1\): \[ (1)^2(1 - 7)(1 + 7) = 1(-6)(8) = -48 < 0 \]

4. Test a point in \((7, \infty)\), e.g. \(x = 8\): \[ (8)^2(8 - 7)(8 + 7) = 64(1)(15) = 960 > 0 \]

Now we summarize the test results:

• The expression is non-positive in the intervals \((-7, 0)\) and \((0, 7)\).
• We also include the points \(x = -7\), \(x = 0\), and \(x = 7\) since the original inequality is non-strict (i.e., \(\geq 0\)).

Thus, the solution in interval notation is:

\[ [-7, 0] \cup [0, 7] \]

Final Answer:

A. The solutions are the values of x in the interval \([-7, 0] \cup [0, 7]\).