Algebra Examples

$| x^{3} - 10 x^{2} |$

Step 1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x^{3} - 10 x^{2} \geq 0$

Step 2

Solve the inequality.

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Step 2.1

Convert the inequality to an equation.

$x^{3} - 10 x^{2} = 0$

Step 2.2

Factor $x^{2}$ out of $x^{3} - 10 x^{2}$ .

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Step 2.2.1

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

$x^{2} x - 10 x^{2} = 0$

Step 2.2.2

Factor $x^{2}$ out of $- 10 x^{2}$ .

$x^{2} x + x^{2} \cdot - 10 = 0$

Step 2.2.3

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot - 10$ .

$x^{2} (x - 10) = 0$

Step 2.3

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

$x^{2} = 0$

$x - 10 = 0$

Step 2.4

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

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Step 2.4.1

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

$x^{2} = 0$

Step 2.4.2

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

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Step 2.4.2.1

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

$x = \pm \sqrt{0}$

Step 2.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.4.2.2.1

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

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

Step 2.4.2.2.2

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

$x = \pm 0$

Step 2.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 2.5

Set $x - 10$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Step 2.5.2

Add $10$ to both sides of the equation.

$x = 10$

Step 2.6

The final solution is all the values that make $x^{2} (x - 10) = 0$ true.

$x = 0, 10$

Step 2.7

Use each root to create test intervals.

$x < 0$

$0 < x < 10$

$x > 10$

Step 2.8

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

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Step 2.8.1

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

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Step 2.8.1.1

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

$x = - 2$

Step 2.8.1.2

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

${(- 2)}^{3} - 10 {(- 2)}^{2} \geq 0$

Step 2.8.1.3

The left side $- 48$ is less than the right side $0$ , which means that the given statement is false.

False

Step 2.8.2

Test a value on the interval $0 < x < 10$ to see if it makes the inequality true.

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Step 2.8.2.1

Choose a value on the interval $0 < x < 10$ and see if this value makes the original inequality true.

$x = 5$

Step 2.8.2.2

Replace $x$ with $5$ in the original inequality.

${(5)}^{3} - 10 {(5)}^{2} \geq 0$

Step 2.8.2.3

The left side $- 125$ is less than the right side $0$ , which means that the given statement is false.

False

Step 2.8.3

Test a value on the interval $x > 10$ to see if it makes the inequality true.

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Step 2.8.3.1

Choose a value on the interval $x > 10$ and see if this value makes the original inequality true.

$x = 12$

Step 2.8.3.2

Replace $x$ with $12$ in the original inequality.

${(12)}^{3} - 10 {(12)}^{2} \geq 0$

Step 2.8.3.3

The left side $288$ is greater than the right side $0$ , which means that the given statement is always true.

True

Step 2.8.4

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

$x < 0$ False

$0 < x < 10$ False

$x > 10$ True

$x < 0$ False

$0 < x < 10$ False

$x > 10$ True

Step 2.9

The solution consists of all of the true intervals.

$x \geq 10$ or $x = 0$

Step 2.10

Combine the intervals.

$x = 0 or x \geq 10$

Step 3

In the piece where $x^{3} - 10 x^{2}$ is non-negative, remove the absolute value.

$x^{3} - 10 x^{2}$

Step 4

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x^{3} - 10 x^{2} < 0$

Step 5

Solve the inequality.

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Step 5.1

Convert the inequality to an equation.

$x^{3} - 10 x^{2} = 0$

Step 5.2

Factor $x^{2}$ out of $x^{3} - 10 x^{2}$ .

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Step 5.2.1

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

$x^{2} x - 10 x^{2} = 0$

Step 5.2.2

Factor $x^{2}$ out of $- 10 x^{2}$ .

$x^{2} x + x^{2} \cdot - 10 = 0$

Step 5.2.3

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot - 10$ .

$x^{2} (x - 10) = 0$

Step 5.3

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

$x^{2} = 0$

$x - 10 = 0$

Step 5.4

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

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Step 5.4.1

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

$x^{2} = 0$

Step 5.4.2

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

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Step 5.4.2.1

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

$x = \pm \sqrt{0}$

Step 5.4.2.2

Simplify $\pm \sqrt{0}$ .

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Step 5.4.2.2.1

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

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

Step 5.4.2.2.2

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

$x = \pm 0$

Step 5.4.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 5.5

Set $x - 10$ equal to $0$ and solve for $x$ .

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Step 5.5.1

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Step 5.5.2

Add $10$ to both sides of the equation.

$x = 10$

Step 5.6

The final solution is all the values that make $x^{2} (x - 10) = 0$ true.

$x = 0, 10$

Step 5.7

Use each root to create test intervals.

$x < 0$

$0 < x < 10$

$x > 10$

Step 5.8

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

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Step 5.8.1

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

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Step 5.8.1.1

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

$x = - 2$

Step 5.8.1.2

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

${(- 2)}^{3} - 10 {(- 2)}^{2} < 0$

Step 5.8.1.3

The left side $- 48$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 5.8.2

Test a value on the interval $0 < x < 10$ to see if it makes the inequality true.

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Step 5.8.2.1

Choose a value on the interval $0 < x < 10$ and see if this value makes the original inequality true.

$x = 5$

Step 5.8.2.2

Replace $x$ with $5$ in the original inequality.

${(5)}^{3} - 10 {(5)}^{2} < 0$

Step 5.8.2.3

The left side $- 125$ is less than the right side $0$ , which means that the given statement is always true.

True

Step 5.8.3

Test a value on the interval $x > 10$ to see if it makes the inequality true.

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Step 5.8.3.1

Choose a value on the interval $x > 10$ and see if this value makes the original inequality true.

$x = 12$

Step 5.8.3.2

Replace $x$ with $12$ in the original inequality.

${(12)}^{3} - 10 {(12)}^{2} < 0$

Step 5.8.3.3

The left side $288$ is not less than the right side $0$ , which means that the given statement is false.

False

Step 5.8.4

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

$x < 0$ True

$0 < x < 10$ True

$x > 10$ False

$x < 0$ True

$0 < x < 10$ True

$x > 10$ False

Step 5.9

The solution consists of all of the true intervals.

$x < 0$ or $0 < x < 10$

Step 6

In the piece where $x^{3} - 10 x^{2}$ is negative, remove the absolute value and multiply by $- 1$ .

$- (x^{3} - 10 x^{2})$

Step 7

Write as a piecewise.

${\begin{cases} x^{3} - 10 x^{2} & x = 0 or x \geq 10 \\ - (x^{3} - 10 x^{2}) & x < 0 or 0 < x < 10 \end{cases}$

Step 8

Simplify $- (x^{3} - 10 x^{2})$ .

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Step 8.1

Apply the distributive property.

${\begin{cases} x^{3} - 10 x^{2} & x = 0 or x \geq 10 \\ - x^{3} - (- 10 x^{2}) & x < 0 or 0 < x < 10 \end{cases}$

Step 8.2

Multiply $- 10$ by $- 1$ .

${\begin{cases} x^{3} - 10 x^{2} & x = 0 or x \geq 10 \\ - x^{3} + 10 x^{2} & x < 0 or 0 < x < 10 \end{cases}$