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

Question

Solve the inequality

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

Alternative Form

x \in (2, \frac{5}{2})

Evaluate

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

Find the domain

More Steps

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

Move the expression to the left side

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

Subtract the terms

More Steps

\frac{10 - 4 x}{x - 2} > 0

Set the numerator and denominator of \frac{10 - 4 x}{x - 2} equal to 0 to find the values of x where sign changes may occur

10 - 4 x = 0 x - 2 = 0

Calculate

More Steps

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

Calculate

More Steps

x = \frac{5}{2} x = 2

Determine the test intervals using the critical values

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

Choose a value form each interval

x_{1} = 1 x_{2} = \frac{9}{4} x_{3} = 4

To determine if x < 2 is the solution to the inequality,test if the chosen value x = 1 satisfies the initial inequality

More Steps

x < 2 is not a solution x_{2} = \frac{9}{4} x_{3} = 4

To determine if 2 < x < \frac{5}{2} is the solution to the inequality,test if the chosen value x = \frac{9}{4} satisfies the initial inequality

More Steps

x < 2 is not a solution 2 < x < \frac{5}{2} is the solution x_{3} = 4

To determine if x > \frac{5}{2} is the solution to the inequality,test if the chosen value x = 4 satisfies the initial inequality

More Steps

x < 2 is not a solution 2 < x < \frac{5}{2} is the solution x > \frac{5}{2} is not a solution

The original inequality is a strict inequality,so does not include the critical value ,the final solution is 2 < x < \frac{5}{2}

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

Check if the solution is in the defined range

2 < x < \frac{5}{2}, x \neq = 2

Solution

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

Alternative Form

x \in (2, \frac{5}{2})

Show Solution