Solve for r |r+5|>20

$| r + 5 | > 20$

Write $| r + 5 | > 20$ as a piecewise.

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

$r + 5 \geq 0$

Subtract $5$ from both sides of the inequality.

$r \geq - 5$

In the piece where $r + 5$ is non-negative, remove the absolute value.

$r + 5 > 20$

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

$r + 5 < 0$

Subtract $5$ from both sides of the inequality.

$r < - 5$

In the piece where $r + 5$ is negative, remove the absolute value and multiply by $- 1$ .

$- (r + 5) > 20$

Write as a piecewise.

${\begin{cases} r + 5 > 20 & r \geq - 5 \\ - (r + 5) > 20 & r < - 5 \end{cases}$

Simplify $- (r + 5) > 20$ .

Apply the distributive property.

${\begin{cases} r + 5 > 20 & r \geq - 5 \\ - r - 1 \cdot 5 > 20 & r < - 5 \end{cases}$

Multiply $- 1$ by $5$ .

${\begin{cases} r + 5 > 20 & r \geq - 5 \\ - r - 5 > 20 & r < - 5 \end{cases}$

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

Subtract $5$ from both sides of the inequality.

$r > 20 - 5$

Subtract $5$ from $20$ .

$r > 15$

Solve $- r - 5 > 20$ for $r$ .

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

Add $5$ to both sides of the inequality.

$- r > 20 + 5$

Add $20$ and $5$ .

$- r > 25$

Divide each term in $- r > 25$ by $- 1$ and simplify.

Divide each term in $- r > 25$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- r}{- 1} < \frac{25}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{r}{1} < \frac{25}{- 1}$

Divide $r$ by $1$ .

$r < \frac{25}{- 1}$

Simplify the right side.

Divide $25$ by $- 1$ .

$r < - 25$

Find the union of the solutions.

$r < - 25$ or $r > 15$

The result can be shown in multiple forms.

Inequality Form:

$r < - 25 or r > 15$

Interval Notation:

$(- \infty, - 25) \cup (15, \infty)$

Solve for r |r+5|>20

Related Questions

Support LCMGCF.com by disabling your adblocker.