Solve (x-4)/(2x-1)>2

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

- \frac{2}{3} < x < \frac{1}{2}

Alternative Form

x \in (- \frac{2}{3}, \frac{1}{2})

Evaluate

\frac{x - 4}{2 x - 1} > 2

Find the domain

More Steps

\frac{x - 4}{2 x - 1} > 2, x \neq = \frac{1}{2}

Move the expression to the left side

\frac{x - 4}{2 x - 1} - 2 > 0

Subtract the terms

More Steps

- \frac{3 x + 2}{2 x - 1} > 0

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

\frac{3 x + 2}{2 x - 1} < 0

Set the numerator and denominator of \frac{3 x + 2}{2 x - 1} equal to 0 to find the values of x where sign changes may occur

3 x + 2 = 0 2 x - 1 = 0

Calculate

More Steps

x = - \frac{2}{3} 2 x - 1 = 0

Calculate

More Steps

x = - \frac{2}{3} x = \frac{1}{2}

Determine the test intervals using the critical values

x < - \frac{2}{3} - \frac{2}{3} < x < \frac{1}{2} x > \frac{1}{2}

Choose a value form each interval

x_{1} = - 2 x_{2} = 0 x_{3} = 2

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

More Steps

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

To determine if - \frac{2}{3} < x < \frac{1}{2} is the solution to the inequality,test if the chosen value x = 0 satisfies the initial inequality

More Steps

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

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

More Steps

x < - \frac{2}{3} is not a solution - \frac{2}{3} < x < \frac{1}{2} is the solution x > \frac{1}{2} is not a solution

The original inequality is a strict inequality,so does not include the critical value ,the final solution is - \frac{2}{3} < x < \frac{1}{2}

- \frac{2}{3} < x < \frac{1}{2}

Check if the solution is in the defined range

- \frac{2}{3} < x < \frac{1}{2}, x \neq = \frac{1}{2}

Solution

- \frac{2}{3} < x < \frac{1}{2}

Alternative Form

x \in (- \frac{2}{3}, \frac{1}{2})

Show Solution