Solve |\frac{x}{\:x-2}|\ge 2

Question

Solve the inequality

x \in [\frac{4}{3}, 2) \cup (2, 4]

Evaluate

\frac{x}{x - 2} \geq 2

Find the domain

More Steps

Evaluate

x - 2 \neq = 0

Move the constant to the right side

x \neq = 0 + 2

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

x \neq = 2

\frac{x}{x - 2} \geq 2, x \neq = 2

Separate the inequality into 2 possible cases

\frac{x}{x - 2} \geq 2 \frac{x}{x - 2} \leq - 2

Solve the inequality for x

More Steps

Evaluate

\frac{x}{x - 2} \geq 2

Calculate

\frac{x}{x - 2} - 2 \geq 0

Calculate

More Steps

Calculate

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

Reduce fractions to a common denominator

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

Write all numerators above the common denominator

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

Multiply the terms

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

Subtract the terms

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

\frac{- x + 4}{x - 2} \geq 0

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

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

Separate the inequality into 2 possible cases

{x - 4 \geq 0 x - 2 < 0 {x - 4 \leq 0 x - 2 > 0

Solve the inequality

More Steps

Evaluate

x - 4 \geq 0

Move the constant to the right side

x \geq 0 + 4

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

x \geq 4

{x \geq 4 x - 2 < 0 {x - 4 \leq 0 x - 2 > 0

Solve the inequality

More Steps

Evaluate

x - 2 < 0

Move the constant to the right side

x < 0 + 2

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

x < 2

{x \geq 4 x < 2 {x - 4 \leq 0 x - 2 > 0

Solve the inequality

More Steps

Evaluate

x - 4 \leq 0

Move the constant to the right side

x \leq 0 + 4

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

x \leq 4

{x \geq 4 x < 2 {x \leq 4 x - 2 > 0

Solve the inequality

More Steps

Evaluate

x - 2 > 0

Move the constant to the right side

x > 0 + 2

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

x > 2

{x \geq 4 x < 2 {x \leq 4 x > 2

Find the intersection

x \in \emptyset {x \leq 4 x > 2

Find the intersection

x \in \emptyset 2 < x \leq 4

Find the union

2 < x \leq 4

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

Solve the inequality for x

More Steps

Evaluate

\frac{x}{x - 2} \leq - 2

Calculate

\frac{x}{x - 2} - (- 2) \leq 0

Calculate

More Steps

Calculate

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

Calculate

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

Reduce fractions to a common denominator

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

Write all numerators above the common denominator

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

Multiply the terms

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

Add the terms

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

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

Separate the inequality into 2 possible cases

{3 x - 4 \geq 0 x - 2 < 0 {3 x - 4 \leq 0 x - 2 > 0

Solve the inequality

More Steps

Evaluate

3 x - 4 \geq 0

Move the constant to the right side

3 x \geq 0 + 4

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

3 x \geq 4

Divide both sides

\frac{3 x}{3} \geq \frac{4}{3}

Divide the numbers

x \geq \frac{4}{3}

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

Solve the inequality

More Steps

Evaluate

x - 2 < 0

Move the constant to the right side

x < 0 + 2

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

x < 2

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

Solve the inequality

More Steps

Evaluate

3 x - 4 \leq 0

Move the constant to the right side

3 x \leq 0 + 4

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

3 x \leq 4

Divide both sides

\frac{3 x}{3} \leq \frac{4}{3}

Divide the numbers

x \leq \frac{4}{3}

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

Solve the inequality

More Steps

Evaluate

x - 2 > 0

Move the constant to the right side

x > 0 + 2

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

x > 2

{x \geq \frac{4}{3} x < 2 {x \leq \frac{4}{3} x > 2

Find the intersection

\frac{4}{3} \leq x < 2 {x \leq \frac{4}{3} x > 2

Find the intersection

\frac{4}{3} \leq x < 2 x \in \emptyset

Find the union

\frac{4}{3} \leq x < 2

2 < x \leq 4 \frac{4}{3} \leq x < 2

Find the union

x \in [\frac{4}{3}, 2) \cup (2, 4]

Check if the solution is in the defined range

x \in [\frac{4}{3}, 2) \cup (2, 4], x \neq = 2

Solution

x \in [\frac{4}{3}, 2) \cup (2, 4]

Show Solution