Basic Math Examples

$| v + 7 | + 6 > 9$

Step 1

Write $| v + 7 | + 6 > 9$ 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.

$v + 7 \geq 0$

Step 1.2

Subtract $7$ from both sides of the inequality.

$v \geq - 7$

Step 1.3

In the piece where $v + 7$ is non-negative, remove the absolute value.

$v + 7 + 6 > 9$

Step 1.4

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

$v + 7 < 0$

Step 1.5

Subtract $7$ from both sides of the inequality.

$v < - 7$

Step 1.6

In the piece where $v + 7$ is negative, remove the absolute value and multiply by $- 1$ .

$- (v + 7) + 6 > 9$

Step 1.7

Write as a piecewise.

${\begin{cases} v + 7 + 6 > 9 & v \geq - 7 \\ - (v + 7) + 6 > 9 & v < - 7 \end{cases}$

Step 1.8

Add $7$ and $6$ .

${\begin{cases} v + 13 > 9 & v \geq - 7 \\ - (v + 7) + 6 > 9 & v < - 7 \end{cases}$

Step 1.9

Simplify $- (v + 7) + 6 > 9$ .

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Step 1.9.1

Simplify each term.

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Step 1.9.1.1

Apply the distributive property.

${\begin{cases} v + 13 > 9 & v \geq - 7 \\ - v - 1 \cdot 7 + 6 > 9 & v < - 7 \end{cases}$

Step 1.9.1.2

Multiply $- 1$ by $7$ .

${\begin{cases} v + 13 > 9 & v \geq - 7 \\ - v - 7 + 6 > 9 & v < - 7 \end{cases}$

Step 1.9.2

Add $- 7$ and $6$ .

${\begin{cases} v + 13 > 9 & v \geq - 7 \\ - v - 1 > 9 & v < - 7 \end{cases}$

Step 2

Move all terms not containing $v$ to the right side of the inequality.

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Step 2.1

Subtract $13$ from both sides of the inequality.

$v > 9 - 13$

Step 2.2

Subtract $13$ from $9$ .

$v > - 4$

Step 3

Solve $- v - 1 > 9$ for $v$ .

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Step 3.1

Move all terms not containing $v$ to the right side of the inequality.

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Step 3.1.1

Add $1$ to both sides of the inequality.

$- v > 9 + 1$

Step 3.1.2

Add $9$ and $1$ .

$- v > 10$

Step 3.2

Divide each term in $- v > 10$ by $- 1$ and simplify.

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Step 3.2.1

Divide each term in $- v > 10$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- v}{- 1} < \frac{10}{- 1}$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

$\frac{v}{1} < \frac{10}{- 1}$

Step 3.2.2.2

Divide $v$ by $1$ .

$v < \frac{10}{- 1}$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Divide $10$ by $- 1$ .

$v < - 10$

Step 4

Find the union of the solutions.

$v < - 10$ or $v > - 4$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$v < - 10 or v > - 4$

Interval Notation:

$(- \infty, - 10) \cup (- 4, \infty)$

Step 6