Solve the Inequality for x 2x^3-5x^2+2x<0

$2 x^{3} - 5 x^{2} + 2 x < 0$

Step 1

Convert the inequality to an equation.

$2 x^{3} - 5 x^{2} + 2 x = 0$

Step 2

Factor the left side of the equation.

Factor $x$ out of $2 x^{3} - 5 x^{2} + 2 x$ .

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

$x (2 x^{2}) - 5 x^{2} + 2 x = 0$

Factor $x$ out of $- 5 x^{2}$ .

$x (2 x^{2}) + x (- 5 x) + 2 x = 0$

Factor $x$ out of $2 x$ .

$x (2 x^{2}) + x (- 5 x) + x \cdot 2 = 0$

Factor $x$ out of $x (2 x^{2}) + x (- 5 x)$ .

$x (2 x^{2} - 5 x) + x \cdot 2 = 0$

Factor $x$ out of $x (2 x^{2} - 5 x) + x \cdot 2$ .

$x (2 x^{2} - 5 x + 2) = 0$

Factor.

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot 2 = 4$ and whose sum is $b = - 5$ .

Factor $- 5$ out of $- 5 x$ .

$x (2 x^{2} - 5 x + 2) = 0$

Rewrite $- 5$ as $- 1$ plus $- 4$

$x (2 x^{2} + (- 1 - 4) x + 2) = 0$

Apply the distributive property.

$x (2 x^{2} - 1 x - 4 x + 2) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$x ((2 x^{2} - 1 x) - 4 x + 2) = 0$

Factor out the greatest common factor (GCF) from each group.

$x (x (2 x - 1) - 2 (2 x - 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$x ((2 x - 1) (x - 2)) = 0$

Remove unnecessary parentheses.

$x (2 x - 1) (x - 2) = 0$

Step 3

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

$x = 0$

$2 x - 1 = 0$

$x - 2 = 0$

Step 4

Set $x$ equal to $0$ .

$x = 0$

Step 5

Set $2 x - 1$ equal to $0$ and solve for $x$ .

Set $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Solve $2 x - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

$2 x = 1$

Divide each term in $2 x = 1$ by $2$ and simplify.

Divide each term in $2 x = 1$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Step 6

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

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

$x - 2 = 0$

Add $2$ to both sides of the equation.

$x = 2$

Step 7

The final solution is all the values that make $x (2 x - 1) (x - 2) = 0$ true.

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

Step 8

Use each root to create test intervals.

$x < 0$

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

$\frac{1}{2} < x < 2$

$x > 2$

Step 9

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.

$2 {(- 2)}^{3} - 5 {(- 2)}^{2} + 2 (- 2) < 0$

The left side $- 40$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $0 < x < \frac{1}{2}$ to see if it makes the inequality true.

Choose a value on the interval $0 < x < \frac{1}{2}$ and see if this value makes the original inequality true.

$x = 0.25$

Replace $x$ with $0.25$ in the original inequality.

$2 {(0.25)}^{3} - 5 {(0.25)}^{2} + 2 (0.25) < 0$

The left side $0.21875$ is not less than the right side $0$ , which means that the given statement is false.

False

Test a value on the interval $\frac{1}{2} < x < 2$ to see if it makes the inequality true.

Choose a value on the interval $\frac{1}{2} < x < 2$ and see if this value makes the original inequality true.

$x = 1$

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

$2 {(1)}^{3} - 5 {(1)}^{2} + 2 (1) < 0$

The left side $- 1$ is less than the right side $0$ , which means that the given statement is always true.

True

Test a value on the interval $x > 2$ to see if it makes the inequality true.

Choose a value on the interval $x > 2$ and see if this value makes the original inequality true.

$x = 4$

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

$2 {(4)}^{3} - 5 {(4)}^{2} + 2 (4) < 0$

The left side $56$ 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

$0 < x < \frac{1}{2}$ False

$\frac{1}{2} < x < 2$ True

$x > 2$ False

$x < 0$ True

$0 < x < \frac{1}{2}$ False

$\frac{1}{2} < x < 2$ True

$x > 2$ False

Step 10

The solution consists of all of the true intervals.

$x < 0$ or $\frac{1}{2} < x < 2$

Step 11

The result can be shown in multiple forms.

Inequality Form:

$x < 0 or \frac{1}{2} < x < 2$

Interval Notation:

$(- \infty, 0) \cup (\frac{1}{2}, 2)$

Step 12

Solve the Inequality for x 2x^3-5x^2+2x<0

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