Solve 1/x 1/3 < 1/5

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

x \in (- \infty, 0) \cup (\frac{5}{3}, + \infty)

Evaluate

\frac{1}{x} \times \frac{1}{3} < \frac{1}{5}

Find the domain

\frac{1}{x} \times \frac{1}{3} < \frac{1}{5}, x \neq = 0

Multiply the terms

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\frac{1}{3 x} < \frac{1}{5}

Move the expression to the left side

\frac{1}{3 x} - \frac{1}{5} < 0

Subtract the terms

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\frac{5 - 3 x}{15 x} < 0

Set the numerator and denominator of \frac{5 - 3 x}{15 x} equal to 0 to find the values of x where sign changes may occur

5 - 3 x = 0 15 x = 0

Calculate

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x = \frac{5}{3} 15 x = 0

Calculate

x = \frac{5}{3} x = 0

Determine the test intervals using the critical values

x < 0 0 < x < \frac{5}{3} x > \frac{5}{3}

Choose a value form each interval

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

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

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x < 0 is the solution x_{2} = 1 x_{3} = 3

To determine if 0 < x < \frac{5}{3} is the solution to the inequality,test if the chosen value x = 1 satisfies the initial inequality

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x < 0 is the solution 0 < x < \frac{5}{3} is not a solution x_{3} = 3

To determine if x > \frac{5}{3} is the solution to the inequality,test if the chosen value x = 3 satisfies the initial inequality

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x < 0 is the solution 0 < x < \frac{5}{3} is not a solution x > \frac{5}{3} is the solution

The original inequality is a strict inequality,so does not include the critical value ,the final solution is x \in (- \infty, 0) \cup (\frac{5}{3}, + \infty)

x \in (- \infty, 0) \cup (\frac{5}{3}, + \infty)

Check if the solution is in the defined range

x \in (- \infty, 0) \cup (\frac{5}{3}, + \infty), x \neq = 0

Solution

x \in (- \infty, 0) \cup (\frac{5}{3}, + \infty)

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