Solve (x^3)(x ^ 2 - 4x^4) <= 0

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

x \in [- \frac{1}{2}, 0] \cup [\frac{1}{2}, + \infty)

Evaluate

x^{3} (x^{2} - 4 x^{4}) \leq 0

Rewrite the expression

x^{3} (x^{2} - 4 x^{4}) = 0

Separate the equation into 2 possible cases

x^{3} = 0 x^{2} - 4 x^{4} = 0

The only way a power can be 0 is when the base equals 0

x = 0 x^{2} - 4 x^{4} = 0

Solve the equation

More Steps

x = 0 x = 0 x = \frac{1}{2} x = - \frac{1}{2}

Find the union

x = 0 x = \frac{1}{2} x = - \frac{1}{2}

Determine the test intervals using the critical values

x < - \frac{1}{2} - \frac{1}{2} < x < 0 0 < x < \frac{1}{2} x > \frac{1}{2}

Choose a value form each interval

x_{1} = - 2 x_{2} = - \frac{1}{4} x_{3} = \frac{1}{4} x_{4} = 2

To determine if x < - \frac{1}{2} is the solution to the inequality,test if the chosen value x = - 2 satisfies the initial inequality

More Steps

x < - \frac{1}{2} is not a solution x_{2} = - \frac{1}{4} x_{3} = \frac{1}{4} x_{4} = 2

To determine if - \frac{1}{2} < x < 0 is the solution to the inequality,test if the chosen value x = - \frac{1}{4} satisfies the initial inequality

More Steps

x < - \frac{1}{2} is not a solution - \frac{1}{2} < x < 0 is the solution x_{3} = \frac{1}{4} x_{4} = 2

To determine if 0 < x < \frac{1}{2} is the solution to the inequality,test if the chosen value x = \frac{1}{4} satisfies the initial inequality

More Steps

x < - \frac{1}{2} is not a solution - \frac{1}{2} < x < 0 is the solution 0 < x < \frac{1}{2} is not a solution x_{4} = 2

To determine if x > \frac{1}{2} is the solution to the inequality,test if the chosen value x = 2 satisfies the initial inequality

More Steps

x < - \frac{1}{2} is not a solution - \frac{1}{2} < x < 0 is the solution 0 < x < \frac{1}{2} is not a solution x > \frac{1}{2} is the solution

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

- \frac{1}{2} \leq x \leq 0 is the solution x \geq \frac{1}{2} is the solution

Solution

x \in [- \frac{1}{2}, 0] \cup [\frac{1}{2}, + \infty)

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