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

$| n | + 2 \geq 5$

Step 1

Write $| n | + 2 \geq 5$ as a piecewise.

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

$n \geq 0$

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

$n + 2 \geq 5$

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

$n < 0$

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

$- n + 2 \geq 5$

Write as a piecewise.

${\begin{cases} n + 2 \geq 5 & n \geq 0 \\ - n + 2 \geq 5 & n < 0 \end{cases}$

Step 2

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

Subtract $2$ from both sides of the inequality.

$n \geq 5 - 2$

Subtract $2$ from $5$ .

$n \geq 3$

Step 3

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

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

Subtract $2$ from both sides of the inequality.

$- n \geq 5 - 2$

Subtract $2$ from $5$ .

$- n \geq 3$

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $n$ by $1$ .

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

Simplify the right side.

Divide $3$ by $- 1$ .

$n \leq - 3$

Step 4

Find the union of the solutions.

$n \leq - 3$ or $n \geq 3$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$n \leq - 3 or n \geq 3$

Interval Notation:

$(- \infty, - 3] \cup [3, \infty)$

Step 6

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

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