Solve (2x-1)(x^3)(x-5)>0

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

x \in (0, \frac{1}{2}) \cup (5, + \infty)

Evaluate

(2 x - 1) x^{3} (x - 5) > 0

Multiply the first two terms

x^{3} (2 x - 1) (x - 5) > 0

Rewrite the expression

x^{3} (2 x - 1) (x - 5) = 0

Separate the equation into 3 possible cases

x^{3} = 0 2 x - 1 = 0 x - 5 = 0

The only way a power can be 0 is when the base equals 0

x = 0 2 x - 1 = 0 x - 5 = 0

Solve the equation

More Steps

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

Solve the equation

More Steps

x = 0 x = \frac{1}{2} x = 5

Determine the test intervals using the critical values

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

Choose a value form each interval

x_{1} = - 1 x_{2} = \frac{1}{4} x_{3} = 3 x_{4} = 6

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} = \frac{1}{4} x_{3} = 3 x_{4} = 6

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 not a solution 0 < x < \frac{1}{2} is the solution x_{3} = 3 x_{4} = 6

To determine if \frac{1}{2} < x < 5 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 < \frac{1}{2} is the solution \frac{1}{2} < x < 5 is not a solution x_{4} = 6

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

More Steps

x < 0 is not a solution 0 < x < \frac{1}{2} is the solution \frac{1}{2} < x < 5 is not a solution x > 5 is the solution

Solution

x \in (0, \frac{1}{2}) \cup (5, + \infty)

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