Solve |(x-2)/x| >4

Question

Solve the inequality

x \in (- \frac{2}{3}, 0) \cup (0, \frac{2}{5})

Evaluate

\frac{x - 2}{x} > 4

Find the domain

\frac{x - 2}{x} > 4, x \neq = 0

Separate the inequality into 2 possible cases

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

Solve the inequality for x

More Steps

Evaluate

\frac{x - 2}{x} > 4

Calculate

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

Calculate

More Steps

Calculate

\frac{x - 2}{x} - 4

Reduce fractions to a common denominator

\frac{x - 2}{x} - \frac{4 x}{x}

Write all numerators above the common denominator

\frac{x - 2 - 4 x}{x}

Subtract the terms

\frac{- 3 x - 2}{x}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

- \frac{3 x + 2}{x}

- \frac{3 x + 2}{x} > 0

Change the signs on both sides of the inequality and flip the inequality sign

\frac{3 x + 2}{x} < 0

Separate the inequality into 2 possible cases

{3 x + 2 > 0 x < 0 {3 x + 2 < 0 x > 0

Solve the inequality

More Steps

Evaluate

3 x + 2 > 0

Move the constant to the right side

3 x > 0 - 2

Removing 0 doesn't change the value,so remove it from the expression

3 x > - 2

Divide both sides

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

Divide the numbers

x > \frac{- 2}{3}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

x > - \frac{2}{3}

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

Solve the inequality

More Steps

Evaluate

3 x + 2 < 0

Move the constant to the right side

3 x < 0 - 2

Removing 0 doesn't change the value,so remove it from the expression

3 x < - 2

Divide both sides

\frac{3 x}{3} < \frac{- 2}{3}

Divide the numbers

x < \frac{- 2}{3}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

x < - \frac{2}{3}

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

Find the intersection

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

Find the intersection

- \frac{2}{3} < x < 0 x \in \emptyset

Find the union

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

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

Solve the inequality for x

More Steps

Evaluate

\frac{x - 2}{x} < - 4

Calculate

\frac{x - 2}{x} - (- 4) < 0

Calculate

More Steps

Calculate

\frac{x - 2}{x} - (- 4)

Calculate

\frac{x - 2}{x} + 4

Reduce fractions to a common denominator

\frac{x - 2}{x} + \frac{4 x}{x}

Write all numerators above the common denominator

\frac{x - 2 + 4 x}{x}

Add the terms

\frac{5 x - 2}{x}

\frac{5 x - 2}{x} < 0

Separate the inequality into 2 possible cases

{5 x - 2 > 0 x < 0 {5 x - 2 < 0 x > 0

Solve the inequality

More Steps

Evaluate

5 x - 2 > 0

Move the constant to the right side

5 x > 0 + 2

Removing 0 doesn't change the value,so remove it from the expression

5 x > 2

Divide both sides

\frac{5 x}{5} > \frac{2}{5}

Divide the numbers

x > \frac{2}{5}

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

Solve the inequality

More Steps

Evaluate

5 x - 2 < 0

Move the constant to the right side

5 x < 0 + 2

Removing 0 doesn't change the value,so remove it from the expression

5 x < 2

Divide both sides

\frac{5 x}{5} < \frac{2}{5}

Divide the numbers

x < \frac{2}{5}

{x > \frac{2}{5} x < 0 {x < \frac{2}{5} x > 0

Find the intersection

x \in \emptyset {x < \frac{2}{5} x > 0

Find the intersection

x \in \emptyset 0 < x < \frac{2}{5}

Find the union

0 < x < \frac{2}{5}

- \frac{2}{3} < x < 0 0 < x < \frac{2}{5}

Find the union

x \in (- \frac{2}{3}, 0) \cup (0, \frac{2}{5})

Check if the solution is in the defined range

x \in (- \frac{2}{3}, 0) \cup (0, \frac{2}{5}), x \neq = 0

Solution

x \in (- \frac{2}{3}, 0) \cup (0, \frac{2}{5})

Show Solution