Solve 5/(x-6) > 3/(x^2)

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

x > 6

Alternative Form

x \in (6, + \infty)

Evaluate

\frac{5}{x - 6} > \frac{3}{x ^{2}}

Find the domain

More Steps

\frac{5}{x - 6} > \frac{3}{x ^{2}}, x \in (- \infty, 0) \cup (0, 6) \cup (6, + \infty)

Move the expression to the left side

\frac{5}{x - 6} - \frac{3}{x ^{2}} > 0

Subtract the terms

More Steps

\frac{5 x ^{2} - 3 x + 18}{( x - 6 ) x ^{2}} > 0

Set the numerator and denominator of \frac{5 x ^{2} - 3 x + 18}{( x - 6 ) x ^{2}} equal to 0 to find the values of x where sign changes may occur

5 x^{2} - 3 x + 18 = 0 (x - 6) x^{2} = 0

Calculate

More Steps

x \in / R (x - 6) x^{2} = 0

Calculate

More Steps

x \in / R x = 6 x = 0

Determine the test intervals using the critical values

x < 0 0 < x < 6 x > 6

Choose a value form each interval

x_{1} = - 1 x_{2} = 3 x_{3} = 7

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 not a solution x_{2} = 3 x_{3} = 7

To determine if 0 < x < 6 is the solution to the inequality,test if the chosen value x = 3 satisfies the initial inequality

More Steps

x < 0 is not a solution 0 < x < 6 is not a solution x_{3} = 7

To determine if x > 6 is the solution to the inequality,test if the chosen value x = 7 satisfies the initial inequality

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x < 0 is not a solution 0 < x < 6 is not a solution x > 6 is the solution

The original inequality is a strict inequality,so does not include the critical value ,the final solution is x > 6

x > 6

Check if the solution is in the defined range

x > 6, x \in (- \infty, 0) \cup (0, 6) \cup (6, + \infty)

Solution

x > 6

Alternative Form

x \in (6, + \infty)

Show Solution