Solve 6(x^(2) 1)-(2x-4)(3x^2)<3(5x^21)

Question

Solve the inequality

Solve the inequality by testing the values in the interval

Solve the inequality by separating into cases

x > \frac{1}{2}

Alternative Form

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

Evaluate

6 (x^{2} \times 1) - (2 x - 4) \times 3 x^{2} < 3 (5 x^{2} \times 1)

Remove the parentheses

6 x^{2} \times 1 - (2 x - 4) \times 3 x^{2} < 3 \times 5 x^{2} \times 1

Simplify

More Steps

6 x^{2} - 3 x^{2} (2 x - 4) < 3 \times 5 x^{2} \times 1

Multiply the terms

More Steps

6 x^{2} - 3 x^{2} (2 x - 4) < 15 x^{2}

Move the expression to the left side

6 x^{2} - 3 x^{2} (2 x - 4) - 15 x^{2} < 0

Subtract the terms

More Steps

3 x^{2} - 6 x^{3} < 0

Rewrite the expression

3 x^{2} - 6 x^{3} = 0

Factor the expression

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

Divide both sides

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

Separate the equation into 2 possible cases

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

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

x = 0 1 - 2 x = 0

Solve the equation

More Steps

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

Determine the test intervals using the critical values

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

Choose a value form each interval

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

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} = 2

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 not a solution x_{3} = 2

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

More Steps

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

Solution

x > \frac{1}{2}

Alternative Form

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

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