Solve |(2x-1)/(x-1)| > 2

Question

Solve the inequality

x \in (\frac{3}{4}, 1) \cup (1, + \infty)

Evaluate

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

Find the domain

More Steps

Evaluate

x - 1 \neq = 0

Move the constant to the right side

x \neq = 0 + 1

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

x \neq = 1

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

Separate the inequality into 2 possible cases

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

Solve the inequality for x

More Steps

Evaluate

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

Calculate

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

Calculate

More Steps

Calculate

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

Reduce fractions to a common denominator

\frac{2 x - 1}{x - 1} - \frac{2 ( x - 1 )}{x - 1}

Write all numerators above the common denominator

\frac{2 x - 1 - 2 ( x - 1 )}{x - 1}

Multiply the terms

\frac{2 x - 1 - ( 2 x - 2 )}{x - 1}

Subtract the terms

\frac{1}{x - 1}

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

Rewrite the expression

x - 1 > 0

Move the constant to the right side

x > 0 + 1

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

x > 1

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

Solve the inequality for x

More Steps

Evaluate

\frac{2 x - 1}{x - 1} < - 2

Calculate

\frac{2 x - 1}{x - 1} - (- 2) < 0

Calculate

More Steps

Calculate

\frac{2 x - 1}{x - 1} - (- 2)

Calculate

\frac{2 x - 1}{x - 1} + 2

Reduce fractions to a common denominator

\frac{2 x - 1}{x - 1} + \frac{2 ( x - 1 )}{x - 1}

Write all numerators above the common denominator

\frac{2 x - 1 + 2 ( x - 1 )}{x - 1}

Multiply the terms

\frac{2 x - 1 + 2 x - 2}{x - 1}

Calculate the sum or difference

\frac{4 x - 3}{x - 1}

\frac{4 x - 3}{x - 1} < 0

Separate the inequality into 2 possible cases

{4 x - 3 > 0 x - 1 < 0 {4 x - 3 < 0 x - 1 > 0

Solve the inequality

More Steps

Evaluate

4 x - 3 > 0

Move the constant to the right side

4 x > 0 + 3

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

4 x > 3

Divide both sides

\frac{4 x}{4} > \frac{3}{4}

Divide the numbers

x > \frac{3}{4}

{x > \frac{3}{4} x - 1 < 0 {4 x - 3 < 0 x - 1 > 0

Solve the inequality

More Steps

Evaluate

x - 1 < 0

Move the constant to the right side

x < 0 + 1

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

x < 1

{x > \frac{3}{4} x < 1 {4 x - 3 < 0 x - 1 > 0

Solve the inequality

More Steps

Evaluate

4 x - 3 < 0

Move the constant to the right side

4 x < 0 + 3

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

4 x < 3

Divide both sides

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

Divide the numbers

x < \frac{3}{4}

{x > \frac{3}{4} x < 1 {x < \frac{3}{4} x - 1 > 0

Solve the inequality

More Steps

Evaluate

x - 1 > 0

Move the constant to the right side

x > 0 + 1

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

x > 1

{x > \frac{3}{4} x < 1 {x < \frac{3}{4} x > 1

Find the intersection

\frac{3}{4} < x < 1 {x < \frac{3}{4} x > 1

Find the intersection

\frac{3}{4} < x < 1 x \in \emptyset

Find the union

\frac{3}{4} < x < 1

x > 1 \frac{3}{4} < x < 1

Find the union

x \in (\frac{3}{4}, 1) \cup (1, + \infty)

Check if the solution is in the defined range

x \in (\frac{3}{4}, 1) \cup (1, + \infty), x \neq = 1

Solution

x \in (\frac{3}{4}, 1) \cup (1, + \infty)

Show Solution