Algebra Examples

$| - 9 v | \leq 54$

Step 1

Write $| - 9 v | \leq 54$ as a piecewise.

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Step 1.1

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

$- 9 v \geq 0$

Step 1.2

Divide each term in $- 9 v \geq 0$ by $- 9$ and simplify.

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Step 1.2.1

Divide each term in $- 9 v \geq 0$ by $- 9$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 9 v}{- 9} \leq \frac{0}{- 9}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $- 9$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{- 9 v}{- 9} \leq \frac{0}{- 9}$

Step 1.2.2.1.2

Divide $v$ by $1$ .

$v \leq \frac{0}{- 9}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $0$ by $- 9$ .

$v \leq 0$

Step 1.3

In the piece where $- 9 v$ is non-negative, remove the absolute value.

$- 9 v \leq 54$

Step 1.4

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

$- 9 v < 0$

Step 1.5

Divide each term in $- 9 v < 0$ by $- 9$ and simplify.

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Step 1.5.1

Divide each term in $- 9 v < 0$ by $- 9$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 9 v}{- 9} > \frac{0}{- 9}$

Step 1.5.2

Simplify the left side.

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Step 1.5.2.1

Cancel the common factor of $- 9$ .

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Step 1.5.2.1.1

Cancel the common factor.

$\frac{- 9 v}{- 9} > \frac{0}{- 9}$

Step 1.5.2.1.2

Divide $v$ by $1$ .

$v > \frac{0}{- 9}$

Step 1.5.3

Simplify the right side.

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Step 1.5.3.1

Divide $0$ by $- 9$ .

$v > 0$

Step 1.6

In the piece where $- 9 v$ is negative, remove the absolute value and multiply by $- 1$ .

$9 v \leq 54$

Step 1.7

Write as a piecewise.

${\begin{cases} - 9 v \leq 54 & v \leq 0 \\ 9 v \leq 54 & v > 0 \end{cases}$

Step 2

Solve $- 9 v \leq 54$ when $v \leq 0$ .

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Step 2.1

Divide each term in $- 9 v \leq 54$ by $- 9$ and simplify.

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Step 2.1.1

Divide each term in $- 9 v \leq 54$ by $- 9$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 9 v}{- 9} \geq \frac{54}{- 9}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $- 9$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{- 9 v}{- 9} \geq \frac{54}{- 9}$

Step 2.1.2.1.2

Divide $v$ by $1$ .

$v \geq \frac{54}{- 9}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Divide $54$ by $- 9$ .

$v \geq - 6$

Step 2.2

Find the intersection of $v \geq - 6$ and $v \leq 0$ .

$- 6 \leq v \leq 0$

Step 3

Solve $9 v \leq 54$ when $v > 0$ .

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Step 3.1

Divide each term in $9 v \leq 54$ by $9$ and simplify.

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Step 3.1.1

Divide each term in $9 v \leq 54$ by $9$ .

$\frac{9 v}{9} \leq \frac{54}{9}$

Step 3.1.2

Simplify the left side.

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Step 3.1.2.1

Cancel the common factor of $9$ .

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Step 3.1.2.1.1

Cancel the common factor.

$\frac{9 v}{9} \leq \frac{54}{9}$

Step 3.1.2.1.2

Divide $v$ by $1$ .

$v \leq \frac{54}{9}$

Step 3.1.3

Simplify the right side.

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Step 3.1.3.1

Divide $54$ by $9$ .

$v \leq 6$

Step 3.2

Find the intersection of $v \leq 6$ and $v > 0$ .

$0 < v \leq 6$

Step 4

Find the union of the solutions.

$- 6 \leq v \leq 6$

Step 5