Solve 3/(x - 2) < 1

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

x \in (- \infty, 2) \cup (5, + \infty)

Evaluate

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

Find the domain

More Steps

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

Move the expression to the left side

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

Subtract the terms

More Steps

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

Set the numerator and denominator of \frac{5 - x}{x - 2} equal to 0 to find the values of x where sign changes may occur

5 - x = 0 x - 2 = 0

Calculate

More Steps

x = 5 x - 2 = 0

Calculate

More Steps

x = 5 x = 2

Determine the test intervals using the critical values

x < 2 2 < x < 5 x > 5

Choose a value form each interval

x_{1} = 1 x_{2} = 4 x_{3} = 6

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 the solution x_{2} = 4 x_{3} = 6

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

More Steps

x < 2 is the solution 2 < x < 5 is not a solution x_{3} = 6

To determine if x > 5 is the solution to the inequality,test if the chosen value x = 6 satisfies the initial inequality

More Steps

x < 2 is the solution 2 < x < 5 is not a solution x > 5 is the solution

The original inequality is a strict inequality,so does not include the critical value ,the final solution is x \in (- \infty, 2) \cup (5, + \infty)

x \in (- \infty, 2) \cup (5, + \infty)

Check if the solution is in the defined range

x \in (- \infty, 2) \cup (5, + \infty), x \neq = 2

Solution

x \in (- \infty, 2) \cup (5, + \infty)

Show Solution