Solve for n -5|-n+7|>=-45

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

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

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.

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

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

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

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

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

Simplify the right side.

Divide $- 45$ by $- 5$ .

$| - n + 7 | \leq 9$

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

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

$- n + 7 \geq 0$

Solve the inequality.

Subtract $7$ from both sides of the inequality.

$- n \geq - 7$

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

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.

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $n$ by $1$ .

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

Simplify the right side.

Divide $- 7$ by $- 1$ .

$n \leq 7$

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

$- n + 7 \leq 9$

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

$- n + 7 < 0$

Solve the inequality.

Subtract $7$ from both sides of the inequality.

$- n < - 7$

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

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.

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $n$ by $1$ .

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

Simplify the right side.

Divide $- 7$ by $- 1$ .

$n > 7$

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

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

Write as a piecewise.

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

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

Apply the distributive property.

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

Multiply $- - n$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $n$ by $1$ .

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

Multiply $- 1$ by $7$ .

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

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

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

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

Subtract $7$ from both sides of the inequality.

$- n \leq 9 - 7$

Subtract $7$ from $9$ .

$- n \leq 2$

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

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.

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $n$ by $1$ .

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

Simplify the right side.

Divide $2$ by $- 1$ .

$n \geq - 2$

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

$- 2 \leq n \leq 7$

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

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

Add $7$ to both sides of the inequality.

$n \leq 9 + 7$

Add $9$ and $7$ .

$n \leq 16$

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

$7 < n \leq 16$

Find the union of the solutions.

$- 2 \leq n \leq 16$

The result can be shown in multiple forms.

Inequality Form:

$- 2 \leq n \leq 16$

Interval Notation:

$[- 2, 16]$

Solve for n -5|-n+7|>=-45

Related Questions

Support LCMGCF.com by disabling your adblocker.