Solve {(x^2) 5}/(x 1)(x-5) >0

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

Solve for x

x \in (- \infty, 0) \cup (5, + \infty)

Evaluate

\frac{x ^{2} \times 5}{x \times 1} \times (x - 5) > 0

Find the domain

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\frac{x ^{2} \times 5}{x \times 1} \times (x - 5) > 0, x \neq = 0

Simplify

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5 x (x - 5) > 0

Rewrite the expression

5 x (x - 5) = 0

Elimination the left coefficient

x (x - 5) = 0

Separate the equation into 2 possible cases

x = 0 x - 5 = 0

Solve the equation

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x = 0 x = 5

Determine the test intervals using the critical values

x < 0 0 < x < 5 x > 5

Choose a value form each interval

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

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

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

To determine if 0 < x < 5 is the solution to the inequality,test if the chosen value x = 3 satisfies the initial inequality

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x < 0 is the solution 0 < 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 < 0 is the solution 0 < 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, 0) \cup (5, + \infty)

x \in (- \infty, 0) \cup (5, + \infty)

Check if the solution is in the defined range

x \in (- \infty, 0) \cup (5, + \infty), x \neq = 0

Solution

x \in (- \infty, 0) \cup (5, + \infty)

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