Solve 1-(1/(x-1)) <= 3

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

x \in (- \infty, \frac{1}{2}] \cup (1, + \infty)

Evaluate

1 - (\frac{1}{x - 1}) \leq 3

Find the domain

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1 - (\frac{1}{x - 1}) \leq 3, x \neq = 1

Remove the unnecessary parentheses

1 - \frac{1}{x - 1} \leq 3

Move the expression to the left side

1 - \frac{1}{x - 1} - 3 \leq 0

Subtract the terms

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

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

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

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

2 x - 1 = 0 x - 1 = 0

Calculate

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x = \frac{1}{2} x - 1 = 0

Calculate

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x = \frac{1}{2} x = 1

Determine the test intervals using the critical values

x < \frac{1}{2} \frac{1}{2} < x < 1 x > 1

Choose a value form each interval

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

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

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

To determine if \frac{1}{2} < x < 1 is the solution to the inequality,test if the chosen value x = \frac{3}{4} satisfies the initial inequality

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x < \frac{1}{2} is the solution \frac{1}{2} < x < 1 is not a 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

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x < \frac{1}{2} is the solution \frac{1}{2} < x < 1 is not a solution x > 1 is the solution

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

x \leq \frac{1}{2} is the solution x > 1 is the solution

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

x \in (- \infty, \frac{1}{2}] \cup (1, + \infty)

Check if the solution is in the defined range

x \in (- \infty, \frac{1}{2}] \cup (1, + \infty), x \neq = 1

Solution

x \in (- \infty, \frac{1}{2}] \cup (1, + \infty)

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