Solve x/(x-5) > 1/2

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

x \in (- \infty, - 5) \cup (5, + \infty)

Evaluate

\frac{x}{x - 5} > \frac{1}{2}

Find the domain

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\frac{x}{x - 5} > \frac{1}{2}, x \neq = 5

Move the expression to the left side

\frac{x}{x - 5} - \frac{1}{2} > 0

Subtract the terms

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\frac{x + 5}{2 ( x - 5 )} > 0

Set the numerator and denominator of \frac{x + 5}{2 ( x - 5 )} equal to 0 to find the values of x where sign changes may occur

x + 5 = 0 2 (x - 5) = 0

Calculate

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x = - 5 2 (x - 5) = 0

Calculate

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x = - 5 x = 5

Determine the test intervals using the critical values

x < - 5 - 5 < x < 5 x > 5

Choose a value form each interval

x_{1} = - 6 x_{2} = 0 x_{3} = 6

To determine if x < - 5 is the solution to the inequality,test if the chosen value x = - 6 satisfies the initial inequality

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x < - 5 is the solution x_{2} = 0 x_{3} = 6

To determine if - 5 < x < 5 is the solution to the inequality,test if the chosen value x = 0 satisfies the initial inequality

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x < - 5 is the solution - 5 < x < 5 is not a solution x_{3} = 6

To determine if x > 5 is the solution to the inequality,test if the chosen value x = 6 satisfies the initial inequality

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x < - 5 is the solution - 5 < x < 5 is not a solution x > 5 is the solution

The original inequality is a strict inequality,so does not include the critical value ,the final solution is x \in (- \infty, - 5) \cup (5, + \infty)

x \in (- \infty, - 5) \cup (5, + \infty)

Check if the solution is in the defined range

x \in (- \infty, - 5) \cup (5, + \infty), x \neq = 5

Solution

x \in (- \infty, - 5) \cup (5, + \infty)

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