Solve for n |n-2|<=5

$| n - 2 | \leq 5$

Step 1

Write $| n - 2 | \leq 5$ 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.

$n - 2 \leq 5$

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 $- 1$ .

$- (n - 2) \leq 5$

Write as a piecewise.

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

Simplify $- (n - 2) \leq 5$ .

Apply the distributive property.

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

Multiply $- 1$ by $- 2$ .

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

Step 2

Solve $n - 2 \leq 5$ when $n \geq 2$ .

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

Add $2$ to both sides of the inequality.

$n \leq 5 + 2$

Add $5$ and $2$ .

$n \leq 7$

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

$2 \leq n \leq 7$

Step 3

Solve $- n + 2 \leq 5$ when $n < 2$ .

Solve $- n + 2 \leq 5$ for $n$ .

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

Subtract $2$ from both sides of the inequality.

$- n \leq 5 - 2$

Subtract $2$ from $5$ .

$- n \leq 3$

Divide each term in $- n \leq 3$ by $- 1$ and simplify.

Divide each term in $- n \leq 3$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- n}{- 1} \geq \frac{3}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{n}{1} \geq \frac{3}{- 1}$

Divide $n$ by $1$ .

$n \geq \frac{3}{- 1}$

Simplify the right side.

Divide $3$ by $- 1$ .

$n \geq - 3$

Find the intersection of $n \geq - 3$ and $n < 2$ .

$- 3 \leq n < 2$

Step 4

Find the union of the solutions.

$- 3 \leq n \leq 7$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 3 \leq n \leq 7$

Interval Notation:

$[- 3, 7]$

Step 6

Solve for n |n-2|<=5

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