Solve x^3>\frac{x}{2}

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

x \in (- \frac{2}{2}, 0) \cup (\frac{2}{2}, + \infty)

Evaluate

x^{3} > \frac{x}{2}

Multiply both sides of the inequality by 2

x^{3} \times 2 > \frac{x}{2} \times 2

Multiply the terms

2 x^{3} > \frac{x}{2} \times 2

Multiply the terms

2 x^{3} > x

Move the expression to the left side

2 x^{3} - x > 0

Rewrite the expression

2 x^{3} - x = 0

Factor the expression

x (2 x^{2} - 1) = 0

Separate the equation into 2 possible cases

x = 0 2 x^{2} - 1 = 0

Solve the equation

More Steps

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

Determine the test intervals using the critical values

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

Choose a value form each interval

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

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

More Steps

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

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

More Steps

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

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

More Steps

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

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

More Steps

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

Solution

x \in (- \frac{2}{2}, 0) \cup (\frac{2}{2}, + \infty)

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