Solve x^3/x-2> 1/2

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve for x

x \in (- \infty, - \frac{10}{2}) \cup (\frac{10}{2}, + \infty)

Evaluate

\frac{x ^{3}}{x} - 2 > \frac{1}{2}

Find the domain

\frac{x ^{3}}{x} - 2 > \frac{1}{2}, x \neq = 0

Divide the terms

More Steps

x^{2} - 2 > \frac{1}{2}

Move the expression to the left side

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

Subtract the numbers

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x^{2} - \frac{5}{2} > 0

Rewrite the expression

x^{2} - \frac{5}{2} = 0

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

x^{2} = 0 + \frac{5}{2}

Add the terms

x^{2} = \frac{5}{2}

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

x = \pm \frac{5}{2}

Simplify the expression

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x = \pm \frac{10}{2}

Separate the equation into 2 possible cases

x = \frac{10}{2} x = - \frac{10}{2}

Determine the test intervals using the critical values

x < - \frac{10}{2} - \frac{10}{2} < x < \frac{10}{2} x > \frac{10}{2}

Choose a value form each interval

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

To determine if x < - \frac{10}{2} is the solution to the inequality,test if the chosen value x = - 3 satisfies the initial inequality

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x < - \frac{10}{2} is the solution x_{2} = 0 x_{3} = 3

To determine if - \frac{10}{2} < x < \frac{10}{2} is the solution to the inequality,test if the chosen value x = 0 satisfies the initial inequality

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x < - \frac{10}{2} is the solution - \frac{10}{2} < x < \frac{10}{2} is not a solution x_{3} = 3

To determine if x > \frac{10}{2} is the solution to the inequality,test if the chosen value x = 3 satisfies the initial inequality

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x < - \frac{10}{2} is the solution - \frac{10}{2} < x < \frac{10}{2} is not a solution x > \frac{10}{2} 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{10}{2}) \cup (\frac{10}{2}, + \infty)

x \in (- \infty, - \frac{10}{2}) \cup (\frac{10}{2}, + \infty)

Check if the solution is in the defined range

x \in (- \infty, - \frac{10}{2}) \cup (\frac{10}{2}, + \infty), x \neq = 0

Solution

x \in (- \infty, - \frac{10}{2}) \cup (\frac{10}{2}, + \infty)

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