Solve x: |x 1| |x| > 3

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

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

Alternative Form

x \in (0, \frac{1}{3})

Evaluate

x \div (∣ x \times 1 ∣ ∣ x ∣) > 3

Find the domain

More Steps

x \div (∣ x \times 1 ∣ ∣ x ∣) > 3, x \neq = 0

Simplify

More Steps

\frac{1}{x} > 3

Move the expression to the left side

\frac{1}{x} - 3 > 0

Subtract the terms

More Steps

\frac{1 - 3 x}{x} > 0

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

1 - 3 x = 0 x = 0

Calculate

More Steps

x = \frac{1}{3} x = 0

Determine the test intervals using the critical values

x < 0 0 < x < \frac{1}{3} x > \frac{1}{3}

Choose a value form each interval

x_{1} = - 1 x_{2} = \frac{1}{6} x_{3} = 2

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 < 0 is not a solution x_{2} = \frac{1}{6} x_{3} = 2

To determine if 0 < x < \frac{1}{3} is the solution to the inequality,test if the chosen value x = \frac{1}{6} satisfies the initial inequality

More Steps

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

To determine if x > \frac{1}{3} is the solution to the inequality,test if the chosen value x = 2 satisfies the initial inequality

More Steps

x < 0 is not a solution 0 < x < \frac{1}{3} is the solution x > \frac{1}{3} is not a solution

The original inequality is a strict inequality,so does not include the critical value ,the final solution is 0 < x < \frac{1}{3}

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

Check if the solution is in the defined range

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

Solution

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

Alternative Form

x \in (0, \frac{1}{3})

Show Solution