Solve the Inequality for x 4x^3-6x^2<0

$4 x^{3} - 6 x^{2} < 0$

Step 1

Convert the inequality to an equation.

$4 x^{3} - 6 x^{2} = 0$

Step 2

Factor $2 x^{2}$ out of $4 x^{3} - 6 x^{2}$ .

Factor $2 x^{2}$ out of $4 x^{3}$ .

$2 x^{2} (2 x) - 6 x^{2} = 0$

Factor $2 x^{2}$ out of $- 6 x^{2}$ .

$2 x^{2} (2 x) + 2 x^{2} (- 3) = 0$

Factor $2 x^{2}$ out of $2 x^{2} (2 x) + 2 x^{2} (- 3)$ .

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

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} = 0$

$2 x - 3 = 0$

Step 4

Set $x^{2}$ equal to $0$ and solve for $x$ .

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Solve $x^{2} = 0$ for $x$ .

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Step 5

Set $2 x - 3$ equal to $0$ and solve for $x$ .

Set $2 x - 3$ equal to $0$ .

$2 x - 3 = 0$

Solve $2 x - 3 = 0$ for $x$ .

Add $3$ to both sides of the equation.

$2 x = 3$

Divide each term in $2 x = 3$ by $2$ and simplify.

Divide each term in $2 x = 3$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Step 6

The final solution is all the values that make $2 x^{2} (2 x - 3) = 0$ true.

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

Step 7

Use each root to create test intervals.

$x < 0$

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

$x > \frac{3}{2}$

Step 8

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Test a value on the interval $x < 0$ to see if it makes the inequality true.

Choose a value on the interval $x < 0$ and see if this value makes the original inequality true.

$x = - 2$

Replace $x$ with $- 2$ in the original inequality.

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

The left side $- 56$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $0 < x < \frac{3}{2}$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < \frac{3}{2}$ and see if this value makes the original inequality true.

$x = 1$

Replace $x$ with $1$ in the original inequality.

$4 {(1)}^{3} - 6 {(1)}^{2} < 0$

The left side $- 2$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > \frac{3}{2}$ to see if it makes the inequality true.

Choose a value on the interval $x > \frac{3}{2}$ and see if this value makes the original inequality true.

$x = 4$

Replace $x$ with $4$ in the original inequality.

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

The left side $160$ is not less than the right side $0$ , which means that the given statement is false.

False

Compare the intervals to determine which ones satisfy the original inequality.

$x < 0$ True

$0 < x < \frac{3}{2}$ True

$x > \frac{3}{2}$ False

$x < 0$ True

$0 < x < \frac{3}{2}$ True

$x > \frac{3}{2}$ False

Step 9

The solution consists of all of the true intervals.

$x < 0$ or $0 < x < \frac{3}{2}$

Step 10

The result can be shown in multiple forms.

Inequality Form:

$x < 0 or 0 < x < \frac{3}{2}$

Interval Notation:

$(- \infty, 0) \cup (0, \frac{3}{2})$

Step 11

Solve the Inequality for x 4x^3-6x^2<0

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