Solve the Absolute Value Inequality for n 5|9-5n|-7=38

$5 | 9 - 5 n | - 7 = 38$

Step 1

Move all terms not containing $| 9 - 5 n |$ to the right side of the equation.

Add $7$ to both sides of the equation.

$5 | 9 - 5 n | = 38 + 7$

Add $38$ and $7$ .

$5 | 9 - 5 n | = 45$

Step 2

Divide each term in $5 | 9 - 5 n | = 45$ by $5$ and simplify.

Divide each term in $5 | 9 - 5 n | = 45$ by $5$ .

$\frac{5 | 9 - 5 n |}{5} = \frac{45}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 | 9 - 5 n |}{5} = \frac{45}{5}$

Divide $| 9 - 5 n |$ by $1$ .

$| 9 - 5 n | = \frac{45}{5}$

Simplify the right side.

Divide $45$ by $5$ .

$| 9 - 5 n | = 9$

Step 3

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$9 - 5 n = \pm 9$

Step 4

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$9 - 5 n = 9$

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

Subtract $9$ from both sides of the equation.

$- 5 n = 9 - 9$

Subtract $9$ from $9$ .

$- 5 n = 0$

Divide each term in $- 5 n = 0$ by $- 5$ and simplify.

Divide each term in $- 5 n = 0$ by $- 5$ .

$\frac{- 5 n}{- 5} = \frac{0}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 n}{- 5} = \frac{0}{- 5}$

Divide $n$ by $1$ .

$n = \frac{0}{- 5}$

Simplify the right side.

Divide $0$ by $- 5$ .

$n = 0$

Next, use the negative value of the $\pm$ to find the second solution.

$9 - 5 n = - 9$

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

Subtract $9$ from both sides of the equation.

$- 5 n = - 9 - 9$

Subtract $9$ from $- 9$ .

$- 5 n = - 18$

Divide each term in $- 5 n = - 18$ by $- 5$ and simplify.

Divide each term in $- 5 n = - 18$ by $- 5$ .

$\frac{- 5 n}{- 5} = \frac{- 18}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 n}{- 5} = \frac{- 18}{- 5}$

Divide $n$ by $1$ .

$n = \frac{- 18}{- 5}$

Simplify the right side.

Dividing two negative values results in a positive value.

$n = \frac{18}{5}$

The complete solution is the result of both the positive and negative portions of the solution.

$n = 0, \frac{18}{5}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$n = 0, \frac{18}{5}$

Decimal Form:

$n = 0, 3.6$

Mixed Number Form:

$n = 0, 3 \frac{3}{5}$

Solve the Absolute Value Inequality for n 5|9-5n|-7=38

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