Solve 15/(x^2) > 6/(x-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, 0) \cup (0, 2)

Evaluate

\frac{15}{x ^{2}} > \frac{6}{x - 2}

Find the domain

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\frac{15}{x ^{2}} > \frac{6}{x - 2}, x \in (- \infty, 0) \cup (0, 2) \cup (2, + \infty)

Move the expression to the left side

\frac{15}{x ^{2}} - \frac{6}{x - 2} > 0

Subtract the terms

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\frac{15 x - 30 - 6 x ^{2}}{x ^{2} ( x - 2 )} > 0

Set the numerator and denominator of \frac{15 x - 30 - 6 x ^{2}}{x ^{2} ( x - 2 )} equal to 0 to find the values of x where sign changes may occur

15 x - 30 - 6 x^{2} = 0 x^{2} (x - 2) = 0

Calculate

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x \in / R x^{2} (x - 2) = 0

Calculate

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x \in / R x = 0 x = 2

Determine the test intervals using the critical values

x < 0 0 < x < 2 x > 2

Choose a value form each interval

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

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} = 1 x_{3} = 3

To determine if 0 < x < 2 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 0 < x < 2 is the solution x_{3} = 3

To determine if x > 2 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 < 2 is the solution x > 2 is not a solution

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

x \in (- \infty, 0) \cup (0, 2)

Check if the solution is in the defined range

x \in (- \infty, 0) \cup (0, 2), x \in (- \infty, 0) \cup (0, 2) \cup (2, + \infty)

Solution

x \in (- \infty, 0) \cup (0, 2)

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