Solve 2/(x-1) > 7/(x 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, 0) \cup (1, \frac{7}{5})

Evaluate

\frac{2}{x - 1} > \frac{7}{x \times 1}

Find the domain

More Steps

\frac{2}{x - 1} > \frac{7}{x \times 1}, x \in (- \infty, 0) \cup (0, 1) \cup (1, + \infty)

Any expression multiplied by 1 remains the same

\frac{2}{x - 1} > \frac{7}{x}

Move the expression to the left side

\frac{2}{x - 1} - \frac{7}{x} > 0

Subtract the terms

More Steps

\frac{- 5 x + 7}{( x - 1 ) x} > 0

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

\frac{5 x - 7}{( x - 1 ) x} < 0

Set the numerator and denominator of \frac{5 x - 7}{( x - 1 ) x} equal to 0 to find the values of x where sign changes may occur

5 x - 7 = 0 (x - 1) x = 0

Calculate

More Steps

x = \frac{7}{5} (x - 1) x = 0

Calculate

More Steps

x = \frac{7}{5} x = 1 x = 0

Determine the test intervals using the critical values

x < 0 0 < x < 1 1 < x < \frac{7}{5} x > \frac{7}{5}

Choose a value form each interval

x_{1} = - 1 x_{2} = \frac{1}{2} x_{3} = \frac{6}{5} x_{4} = 2

To determine if x < 0 is the solution to the inequality,test if the chosen value x = - 1 satisfies the initial inequality

More Steps

x < 0 is the solution x_{2} = \frac{1}{2} x_{3} = \frac{6}{5} x_{4} = 2

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

More Steps

x < 0 is the solution 0 < x < 1 is not a solution x_{3} = \frac{6}{5} x_{4} = 2

To determine if 1 < x < \frac{7}{5} is the solution to the inequality,test if the chosen value x = \frac{6}{5} satisfies the initial inequality

More Steps

x < 0 is the solution 0 < x < 1 is not a solution 1 < x < \frac{7}{5} is the solution x_{4} = 2

To determine if x > \frac{7}{5} is the solution to the inequality,test if the chosen value x = 2 satisfies the initial inequality

More Steps

x < 0 is the solution 0 < x < 1 is not a solution 1 < x < \frac{7}{5} is the solution x > \frac{7}{5} is not a solution

The original inequality is a strict inequality,so does not include the critical value ,the final solution is x \in (- \infty, 0) \cup (1, \frac{7}{5})

x \in (- \infty, 0) \cup (1, \frac{7}{5})

Check if the solution is in the defined range

x \in (- \infty, 0) \cup (1, \frac{7}{5}), x \in (- \infty, 0) \cup (0, 1) \cup (1, + \infty)

Solution

x \in (- \infty, 0) \cup (1, \frac{7}{5})

Show Solution