Algebra Examples

$- 5 | - n + 7 | \geq - 45$

Step 1

Divide each term in $- 5 | - n + 7 | \geq - 45$ by $- 5$ and simplify.

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Step 1.1

Divide each term in $- 5 | - n + 7 | \geq - 45$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 | - n + 7 |}{- 5} \leq \frac{- 45}{- 5}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $- 5$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{- 5 | - n + 7 |}{- 5} \leq \frac{- 45}{- 5}$

Step 1.2.1.2

Divide $| - n + 7 |$ by $1$ .

$| - n + 7 | \leq \frac{- 45}{- 5}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Divide $- 45$ by $- 5$ .

$| - n + 7 | \leq 9$

Step 2

Write $| - n + 7 | \leq 9$ as a piecewise.

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Step 2.1

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

$- n + 7 \geq 0$

Step 2.2

Solve the inequality.

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Step 2.2.1

Subtract $7$ from both sides of the inequality.

$- n \geq - 7$

Step 2.2.2

Divide each term in $- n \geq - 7$ by $- 1$ and simplify.

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Step 2.2.2.1

Divide each term in $- n \geq - 7$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- n}{- 1} \leq \frac{- 7}{- 1}$

Step 2.2.2.2

Simplify the left side.

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Step 2.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{n}{1} \leq \frac{- 7}{- 1}$

Step 2.2.2.2.2

Divide $n$ by $1$ .

$n \leq \frac{- 7}{- 1}$

Step 2.2.2.3

Simplify the right side.

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Step 2.2.2.3.1

Divide $- 7$ by $- 1$ .

$n \leq 7$

Step 2.3

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

$- n + 7 \leq 9$

Step 2.4

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

$- n + 7 < 0$

Step 2.5

Solve the inequality.

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Step 2.5.1

Subtract $7$ from both sides of the inequality.

$- n < - 7$

Step 2.5.2

Divide each term in $- n < - 7$ by $- 1$ and simplify.

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Step 2.5.2.1

Divide each term in $- n < - 7$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- n}{- 1} > \frac{- 7}{- 1}$

Step 2.5.2.2

Simplify the left side.

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Step 2.5.2.2.1

Dividing two negative values results in a positive value.

$\frac{n}{1} > \frac{- 7}{- 1}$

Step 2.5.2.2.2

Divide $n$ by $1$ .

$n > \frac{- 7}{- 1}$

Step 2.5.2.3

Simplify the right side.

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Step 2.5.2.3.1

Divide $- 7$ by $- 1$ .

$n > 7$

Step 2.6

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

$- (- n + 7) \leq 9$

Step 2.7

Write as a piecewise.

${\begin{cases} - n + 7 \leq 9 & n \leq 7 \\ - (- n + 7) \leq 9 & n > 7 \end{cases}$

Step 2.8

Simplify $- (- n + 7) \leq 9$ .

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Step 2.8.1

Apply the distributive property.

${\begin{cases} - n + 7 \leq 9 & n \leq 7 \\ - - n - 1 \cdot 7 \leq 9 & n > 7 \end{cases}$

Step 2.8.2

Multiply $- - n$ .

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Step 2.8.2.1

Multiply $- 1$ by $- 1$ .

${\begin{cases} - n + 7 \leq 9 & n \leq 7 \\ 1 n - 1 \cdot 7 \leq 9 & n > 7 \end{cases}$

Step 2.8.2.2

Multiply $n$ by $1$ .

${\begin{cases} - n + 7 \leq 9 & n \leq 7 \\ n - 1 \cdot 7 \leq 9 & n > 7 \end{cases}$

Step 2.8.3

Multiply $- 1$ by $7$ .

${\begin{cases} - n + 7 \leq 9 & n \leq 7 \\ n - 7 \leq 9 & n > 7 \end{cases}$

Step 3

Solve $- n + 7 \leq 9$ when $n \leq 7$ .

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Step 3.1

Solve $- n + 7 \leq 9$ for $n$ .

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Step 3.1.1

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

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Step 3.1.1.1

Subtract $7$ from both sides of the inequality.

$- n \leq 9 - 7$

Step 3.1.1.2

Subtract $7$ from $9$ .

$- n \leq 2$

Step 3.1.2

Divide each term in $- n \leq 2$ by $- 1$ and simplify.

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Step 3.1.2.1

Divide each term in $- n \leq 2$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- n}{- 1} \geq \frac{2}{- 1}$

Step 3.1.2.2

Simplify the left side.

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Step 3.1.2.2.1

Dividing two negative values results in a positive value.

$\frac{n}{1} \geq \frac{2}{- 1}$

Step 3.1.2.2.2

Divide $n$ by $1$ .

$n \geq \frac{2}{- 1}$

Step 3.1.2.3

Simplify the right side.

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Step 3.1.2.3.1

Divide $2$ by $- 1$ .

$n \geq - 2$

Step 3.2

Find the intersection of $n \geq - 2$ and $n \leq 7$ .

$- 2 \leq n \leq 7$

Step 4

Solve $n - 7 \leq 9$ when $n > 7$ .

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Step 4.1

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

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Step 4.1.1

Add $7$ to both sides of the inequality.

$n \leq 9 + 7$

Step 4.1.2

Add $9$ and $7$ .

$n \leq 16$

Step 4.2

Find the intersection of $n \leq 16$ and $n > 7$ .

$7 < n \leq 16$

Step 5

Find the union of the solutions.

$- 2 \leq n \leq 16$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$- 2 \leq n \leq 16$

Interval Notation:

$[- 2, 16]$

Step 7