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

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

0 < x \leq 1

Alternative Form

x \in (0, 1]

Evaluate

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

Find the domain

More Steps

\frac{x - 2}{- x} \geq 1, x \neq = 0

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

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

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

\frac{x - 2}{x} \leq - 1

Move the expression to the left side

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

Subtract the terms

More Steps

\frac{2 x - 2}{x} \leq 0

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

2 x - 2 = 0 x = 0

Calculate

More Steps

x = 1 x = 0

Determine the test intervals using the critical values

x < 0 0 < x < 1 x > 1

Choose a value form each interval

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

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

More Steps

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

0 < x \leq 1 is the solution

The final solution of the original inequality is 0 < x \leq 1

0 < x \leq 1

Check if the solution is in the defined range

0 < x \leq 1, x \neq = 0

Solution

0 < x \leq 1

Alternative Form

x \in (0, 1]

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