Solve the Absolute Value Inequality for n -3+|n-2|>5

$- 3 + | n - 2 | > 5$

Step 1

Write $- 3 + | n - 2 | > 5$ as a piecewise.

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

$n - 2 \geq 0$

Add $2$ to both sides of the inequality.

$n \geq 2$

In the piece where $n - 2$ is non-negative, remove the absolute value.

$- 3 + n - 2 > 5$

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

$n - 2 < 0$

Add $2$ to both sides of the inequality.

$n < 2$

In the piece where $n - 2$ is negative, remove the absolute value and multiply by $- 1$ .

$- 3 - (n - 2) > 5$

Write as a piecewise.

${\begin{cases} - 3 + n - 2 > 5 & n \geq 2 \\ - 3 - (n - 2) > 5 & n < 2 \end{cases}$

Subtract $2$ from $- 3$ .

${\begin{cases} n - 5 > 5 & n \geq 2 \\ - 3 - (n - 2) > 5 & n < 2 \end{cases}$

Simplify $- 3 - (n - 2) > 5$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} n - 5 > 5 & n \geq 2 \\ - 3 - n - - 2 > 5 & n < 2 \end{cases}$

Multiply $- 1$ by $- 2$ .

${\begin{cases} n - 5 > 5 & n \geq 2 \\ - 3 - n + 2 > 5 & n < 2 \end{cases}$

Add $- 3$ and $2$ .

${\begin{cases} n - 5 > 5 & n \geq 2 \\ - n - 1 > 5 & n < 2 \end{cases}$

Step 2

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

Add $5$ to both sides of the inequality.

$n > 5 + 5$

Add $5$ and $5$ .

$n > 10$

Step 3

Solve $- n - 1 > 5$ for $n$ .

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

Add $1$ to both sides of the inequality.

$- n > 5 + 1$

Add $5$ and $1$ .

$- n > 6$

Divide each term in $- n > 6$ by $- 1$ and simplify.

Divide each term in $- n > 6$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- n}{- 1} < \frac{6}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{n}{1} < \frac{6}{- 1}$

Divide $n$ by $1$ .

$n < \frac{6}{- 1}$

Simplify the right side.

Divide $6$ by $- 1$ .

$n < - 6$

Step 4

Find the union of the solutions.

$n < - 6$ or $n > 10$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$n < - 6 or n > 10$

Interval Notation:

$(- \infty, - 6) \cup (10, \infty)$

Step 6

Solve the Absolute Value Inequality for n -3+|n-2|>5

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