Solve 2/x-1<3 - ScanMath

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 (\frac{1}{2}, + \infty)

Evaluate

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

Find the domain

\frac{2}{x} - 1 < 3, x \neq = 0

Move the expression to the left side

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

Subtract the terms

More Steps

\frac{2 - 4 x}{x} < 0

Set the numerator and denominator of \frac{2 - 4 x}{x} equal to 0 to find the values of x where sign changes may occur

2 - 4 x = 0 x = 0

Calculate

More Steps

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

Determine the test intervals using the critical values

x < 0 0 < x < \frac{1}{2} x > \frac{1}{2}

Choose a value form each interval

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

To determine if 0 < x < \frac{1}{2} is the solution to the inequality,test if the chosen value x = \frac{1}{4} satisfies the initial inequality

More Steps

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

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

The original inequality is a strict inequality,so does not include the critical value ,the final solution is x \in (- \infty, 0) \cup (\frac{1}{2}, + \infty)

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

Check if the solution is in the defined range

x \in (- \infty, 0) \cup (\frac{1}{2}, + \infty), x \neq = 0

Solution

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

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