Solve x^2(-x^2-25) <=25(-x^2-25)

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

Solve for x

x \in (- \infty, - 5] \cup [5, + \infty)

Evaluate

x^{2} (- x^{2} - 25) \leq 25 (- x^{2} - 25)

Move the expression to the left side

x^{2} (- x^{2} - 25) - 25 (- x^{2} - 25) \leq 0

Subtract the terms

More Steps

- x^{4} + 625 \leq 0

Rewrite the expression

- x^{4} + 625 = 0

Move the constant to the right-hand side and change its sign

- x^{4} = 0 - 625

Removing 0 doesn't change the value,so remove it from the expression

- x^{4} = - 625

Change the signs on both sides of the equation

x^{4} = 625

Take the root of both sides of the equation and remember to use both positive and negative roots

x = \pm 4625

Simplify the expression

More Steps

x = \pm 5

Separate the equation into 2 possible cases

x = 5 x = - 5

Determine the test intervals using the critical values

x < - 5 - 5 < x < 5 x > 5

Choose a value form each interval

x_{1} = - 6 x_{2} = 0 x_{3} = 6

To determine if x < - 5 is the solution to the inequality,test if the chosen value x = - 6 satisfies the initial inequality

More Steps

x < - 5 is the solution x_{2} = 0 x_{3} = 6

To determine if - 5 < x < 5 is the solution to the inequality,test if the chosen value x = 0 satisfies the initial inequality

More Steps

x < - 5 is the solution - 5 < x < 5 is not a solution x_{3} = 6

To determine if x > 5 is the solution to the inequality,test if the chosen value x = 6 satisfies the initial inequality

More Steps

x < - 5 is the solution - 5 < x < 5 is not a solution x > 5 is the solution

The original inequality is a nonstrict inequality,so include the critical value in the solution

x \leq - 5 is the solution x \geq 5 is the solution

Solution

x \in (- \infty, - 5] \cup [5, + \infty)

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