Solve the Absolute Value Inequality for n 2-5|10-2n|>=-28

$2 - 5 | 10 - 2 n | \geq - 28$

Write $2 - 5 | 10 - 2 n | \geq - 28$ as a piecewise.

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

$10 - 2 n \geq 0$

Solve the inequality.

Subtract $10$ from both sides of the inequality.

$- 2 n \geq - 10$

Divide each term in $- 2 n \geq - 10$ by $- 2$ and simplify.

Divide each term in $- 2 n \geq - 10$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 n}{- 2} \leq \frac{- 10}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 n}{- 2} \leq \frac{- 10}{- 2}$

Divide $n$ by $1$ .

$n \leq \frac{- 10}{- 2}$

Simplify the right side.

Divide $- 10$ by $- 2$ .

$n \leq 5$

In the piece where $10 - 2 n$ is non-negative, remove the absolute value.

$2 - 5 (10 - 2 n) \geq - 28$

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

$10 - 2 n < 0$

Solve the inequality.

Subtract $10$ from both sides of the inequality.

$- 2 n < - 10$

Divide each term in $- 2 n < - 10$ by $- 2$ and simplify.

Divide each term in $- 2 n < - 10$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 n}{- 2} > \frac{- 10}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 n}{- 2} > \frac{- 10}{- 2}$

Divide $n$ by $1$ .

$n > \frac{- 10}{- 2}$

Simplify the right side.

Divide $- 10$ by $- 2$ .

$n > 5$

In the piece where $10 - 2 n$ is negative, remove the absolute value and multiply by $- 1$ .

$2 - 5 (- (10 - 2 n)) \geq - 28$

Write as a piecewise.

${\begin{cases} 2 - 5 (10 - 2 n) \geq - 28 & n \leq 5 \\ 2 - 5 (- (10 - 2 n)) \geq - 28 & n > 5 \end{cases}$

Simplify $2 - 5 (10 - 2 n) \geq - 28$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 2 - 5 \cdot 10 - 5 (- 2 n) \geq - 28 & n \leq 5 \\ 2 - 5 (- (10 - 2 n)) \geq - 28 & n > 5 \end{cases}$

Multiply $- 5$ by $10$ .

${\begin{cases} 2 - 50 - 5 (- 2 n) \geq - 28 & n \leq 5 \\ 2 - 5 (- (10 - 2 n)) \geq - 28 & n > 5 \end{cases}$

Multiply $- 2$ by $- 5$ .

${\begin{cases} 2 - 50 + 10 n \geq - 28 & n \leq 5 \\ 2 - 5 (- (10 - 2 n)) \geq - 28 & n > 5 \end{cases}$

Subtract $50$ from $2$ .

${\begin{cases} - 48 + 10 n \geq - 28 & n \leq 5 \\ 2 - 5 (- (10 - 2 n)) \geq - 28 & n > 5 \end{cases}$

Simplify $2 - 5 (- (10 - 2 n)) \geq - 28$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} - 48 + 10 n \geq - 28 & n \leq 5 \\ 2 - 5 (- 1 \cdot 10 - (- 2 n)) \geq - 28 & n > 5 \end{cases}$

Multiply $- 1$ by $10$ .

${\begin{cases} - 48 + 10 n \geq - 28 & n \leq 5 \\ 2 - 5 (- 10 - (- 2 n)) \geq - 28 & n > 5 \end{cases}$

Multiply $- 2$ by $- 1$ .

${\begin{cases} - 48 + 10 n \geq - 28 & n \leq 5 \\ 2 - 5 (- 10 + 2 n) \geq - 28 & n > 5 \end{cases}$

Apply the distributive property.

${\begin{cases} - 48 + 10 n \geq - 28 & n \leq 5 \\ 2 - 5 \cdot - 10 - 5 (2 n) \geq - 28 & n > 5 \end{cases}$

Multiply $- 5$ by $- 10$ .

${\begin{cases} - 48 + 10 n \geq - 28 & n \leq 5 \\ 2 + 50 - 5 (2 n) \geq - 28 & n > 5 \end{cases}$

Multiply $2$ by $- 5$ .

${\begin{cases} - 48 + 10 n \geq - 28 & n \leq 5 \\ 2 + 50 - 10 n \geq - 28 & n > 5 \end{cases}$

Add $2$ and $50$ .

${\begin{cases} - 48 + 10 n \geq - 28 & n \leq 5 \\ 52 - 10 n \geq - 28 & n > 5 \end{cases}$

Solve $- 48 + 10 n \geq - 28$ when $n \leq 5$ .

Solve $- 48 + 10 n \geq - 28$ for $n$ .

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

Add $48$ to both sides of the inequality.

$10 n \geq - 28 + 48$

Add $- 28$ and $48$ .

$10 n \geq 20$

Divide each term in $10 n \geq 20$ by $10$ and simplify.

Divide each term in $10 n \geq 20$ by $10$ .

$\frac{10 n}{10} \geq \frac{20}{10}$

Simplify the left side.

Cancel the common factor of $10$ .

Cancel the common factor.

$\frac{10 n}{10} \geq \frac{20}{10}$

Divide $n$ by $1$ .

$n \geq \frac{20}{10}$

Simplify the right side.

Divide $20$ by $10$ .

$n \geq 2$

Find the intersection of $n \geq 2$ and $n \leq 5$ .

$2 \leq n \leq 5$

Solve $52 - 10 n \geq - 28$ when $n > 5$ .

Solve $52 - 10 n \geq - 28$ for $n$ .

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

Subtract $52$ from both sides of the inequality.

$- 10 n \geq - 28 - 52$

Subtract $52$ from $- 28$ .

$- 10 n \geq - 80$

Divide each term in $- 10 n \geq - 80$ by $- 10$ and simplify.

Divide each term in $- 10 n \geq - 80$ by $- 10$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 10 n}{- 10} \leq \frac{- 80}{- 10}$

Simplify the left side.

Cancel the common factor of $- 10$ .

Cancel the common factor.

$\frac{- 10 n}{- 10} \leq \frac{- 80}{- 10}$

Divide $n$ by $1$ .

$n \leq \frac{- 80}{- 10}$

Simplify the right side.

Divide $- 80$ by $- 10$ .

$n \leq 8$

Find the intersection of $n \leq 8$ and $n > 5$ .

$5 < n \leq 8$

Find the union of the solutions.

$2 \leq n \leq 8$

The result can be shown in multiple forms.

Inequality Form:

$2 \leq n \leq 8$

Interval Notation:

$[2, 8]$

Solve the Absolute Value Inequality for n 2-5|10-2n|>=-28

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