Solve 2x ^ 2 + 1/x > 0

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

x \in (- \infty, - \frac{3 4}{2}) \cup (0, + \infty)

Evaluate

2 x^{2} + \frac{1}{x} > 0

Find the domain

2 x^{2} + \frac{1}{x} > 0, x \neq = 0

Rearrange the terms

\frac{2 x ^{3} + 1}{x} > 0

Set the numerator and denominator of \frac{2 x ^{3} + 1}{x} equal to 0 to find the values of x where sign changes may occur

2 x^{3} + 1 = 0 x = 0

Calculate

More Steps

x = - \frac{3 4}{2} x = 0

Determine the test intervals using the critical values

x < - \frac{3 4}{2} - \frac{3 4}{2} < x < 0 x > 0

Choose a value form each interval

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

To determine if x < - \frac{3 4}{2} is the solution to the inequality,test if the chosen value x = - 2 satisfies the initial inequality

More Steps

x < - \frac{3 4}{2} is the solution x_{2} = - \frac{3 4}{4} x_{3} = 1

To determine if - \frac{3 4}{2} < x < 0 is the solution to the inequality,test if the chosen value x = - \frac{3 4}{4} satisfies the initial inequality

More Steps

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

To determine if x > 0 is the solution to the inequality,test if the chosen value x = 1 satisfies the initial inequality

More Steps

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

The original inequality is a strict inequality,so does not include the critical value ,the final solution is x \in (- \infty, - \frac{3 4}{2}) \cup (0, + \infty)

x \in (- \infty, - \frac{3 4}{2}) \cup (0, + \infty)

Check if the solution is in the defined range

x \in (- \infty, - \frac{3 4}{2}) \cup (0, + \infty), x \neq = 0

Solution

x \in (- \infty, - \frac{3 4}{2}) \cup (0, + \infty)

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