Solve (x^5)(2x-3)^5(-x 7)^3(3x 8) < 0

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 (\frac{3}{2}, + \infty)

Evaluate

x^{5} (2 x - 3)^{5} (- x \times 7)^{3} (3 x \times 8) < 0

Remove the parentheses

x^{5} (2 x - 3)^{5} (- x \times 7)^{3} \times 3 x \times 8 < 0

Simplify

More Steps

- 8232 x^{6} (2 x - 3)^{5} x^{3} < 0

Change the signs on both sides of the inequality and flip the inequality sign

8232 x^{6} (2 x - 3)^{5} x^{3} > 0

Rewrite the expression

8232 x^{6} (2 x - 3)^{5} x^{3} = 0

Elimination the left coefficient

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

Separate the equation into 3 possible cases

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

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

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

Solve the equation

More Steps

x = 0 x = \frac{3}{2} x^{3} = 0

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

x = 0 x = \frac{3}{2} x = 0

Find the union

x = 0 x = \frac{3}{2}

Determine the test intervals using the critical values

x < 0 0 < x < \frac{3}{2} x > \frac{3}{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

More Steps

x < 0 is the solution x_{2} = 1 x_{3} = 3

To determine if 0 < x < \frac{3}{2} is the solution to the inequality,test if the chosen value x = 1 satisfies the initial inequality

More Steps

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

To determine if x > \frac{3}{2} is the solution to the inequality,test if the chosen value x = 3 satisfies the initial inequality

More Steps

x < 0 is the solution 0 < x < \frac{3}{2} is not a solution x > \frac{3}{2} is the solution

Solution

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

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