Solve 2/(x 1) >=3/(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, - 2] \cup (0, 1)

Evaluate

\frac{2}{x \times 1} \geq \frac{3}{x - 1}

Find the domain

More Steps

\frac{2}{x \times 1} \geq \frac{3}{x - 1}, x \in (- \infty, 0) \cup (0, 1) \cup (1, + \infty)

Any expression multiplied by 1 remains the same

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

Move the expression to the left side

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

Subtract the terms

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- \frac{x + 2}{x ( x - 1 )} \geq 0

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

\frac{x + 2}{x ( x - 1 )} \leq 0

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

x + 2 = 0 x (x - 1) = 0

Calculate

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x = - 2 x (x - 1) = 0

Calculate

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x = - 2 x = 0 x = 1

Determine the test intervals using the critical values

x < - 2 - 2 < x < 0 0 < x < 1 x > 1

Choose a value form each interval

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

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

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x < - 2 is the solution x_{2} = - 1 x_{3} = \frac{1}{2} x_{4} = 2

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

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

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x < - 2 is the solution - 2 < x < 0 is not a solution 0 < x < 1 is the solution x_{4} = 2

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

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x < - 2 is the solution - 2 < x < 0 is not a solution 0 < x < 1 is the solution x > 1 is not a solution

The original inequality is a nonstrict inequality,so include the critical value in the solution

x \leq - 2 is the solution 0 < x < 1 is the solution

The final solution of the original inequality is x \in (- \infty, - 2] \cup (0, 1)

x \in (- \infty, - 2] \cup (0, 1)

Check if the solution is in the defined range

x \in (- \infty, - 2] \cup (0, 1), x \in (- \infty, 0) \cup (0, 1) \cup (1, + \infty)

Solution

x \in (- \infty, - 2] \cup (0, 1)

Show Solution