Precalculus Examples

$(x^{2}) \cdot | x - 3 |$

Step 1

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

$x - 3 \geq 0$

Step 2

Add $3$ to both sides of the inequality.

$x \geq 3$

Step 3

In the piece where $x - 3$ is non-negative, remove the absolute value.

$x^{2} \cdot (x - 3)$

Step 4

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

$x - 3 < 0$

Step 5

Add $3$ to both sides of the inequality.

$x < 3$

Step 6

In the piece where $x - 3$ is negative, remove the absolute value and multiply by $- 1$ .

$x^{2} \cdot (- (x - 3))$

Step 7

Write as a piecewise.

${\begin{cases} x^{2} \cdot (x - 3) & x \geq 3 \\ x^{2} \cdot (- (x - 3)) & x < 3 \end{cases}$

Step 8

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

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Step 8.1

Apply the distributive property.

${\begin{cases} x^{2} x + x^{2} \cdot - 3 & x \geq 3 \\ x^{2} \cdot (- (x - 3)) & x < 3 \end{cases}$

Step 8.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 8.2.1

Multiply $x^{2}$ by $x$ .

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Step 8.2.1.1

Raise $x$ to the power of $1$ .

${\begin{cases} x^{2} x^{1} + x^{2} \cdot - 3 & x \geq 3 \\ x^{2} \cdot (- (x - 3)) & x < 3 \end{cases}$

Step 8.2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${\begin{cases} x^{2 + 1} + x^{2} \cdot - 3 & x \geq 3 \\ x^{2} \cdot (- (x - 3)) & x < 3 \end{cases}$

Step 8.2.2

Add $2$ and $1$ .

${\begin{cases} x^{3} + x^{2} \cdot - 3 & x \geq 3 \\ x^{2} \cdot (- (x - 3)) & x < 3 \end{cases}$

Step 8.3

Move $- 3$ to the left of $x^{2}$ .

${\begin{cases} x^{3} - 3 x^{2} & x \geq 3 \\ x^{2} \cdot (- (x - 3)) & x < 3 \end{cases}$

Step 9

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

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Step 9.1

Rewrite using the commutative property of multiplication.

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

Step 9.2

Apply the distributive property.

${\begin{cases} x^{3} - 3 x^{2} & x \geq 3 \\ - x^{2} x - x^{2} \cdot - 3 & x < 3 \end{cases}$

Step 9.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 9.3.1

Move $x$ .

${\begin{cases} x^{3} - 3 x^{2} & x \geq 3 \\ - (x \cdot x^{2}) - x^{2} \cdot - 3 & x < 3 \end{cases}$

Step 9.3.2

Multiply $x$ by $x^{2}$ .

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Step 9.3.2.1

Raise $x$ to the power of $1$ .

${\begin{cases} x^{3} - 3 x^{2} & x \geq 3 \\ - (x^{1} x^{2}) - x^{2} \cdot - 3 & x < 3 \end{cases}$

Step 9.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${\begin{cases} x^{3} - 3 x^{2} & x \geq 3 \\ - x^{1 + 2} - x^{2} \cdot - 3 & x < 3 \end{cases}$

Step 9.3.3

Add $1$ and $2$ .

${\begin{cases} x^{3} - 3 x^{2} & x \geq 3 \\ - x^{3} - x^{2} \cdot - 3 & x < 3 \end{cases}$

Step 9.4

Multiply $- 3$ by $- 1$ .

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