Solve for r |5-5r|+1<36

$| 5 - 5 r | + 1 < 36$

Step 1

Write $| 5 - 5 r | + 1 < 36$ as a piecewise.

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

$5 - 5 r \geq 0$

Solve the inequality.

Subtract $5$ from both sides of the inequality.

$- 5 r \geq - 5$

Divide each term in $- 5 r \geq - 5$ by $- 5$ and simplify.

Divide each term in $- 5 r \geq - 5$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 r}{- 5} \leq \frac{- 5}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 r}{- 5} \leq \frac{- 5}{- 5}$

Divide $r$ by $1$ .

$r \leq \frac{- 5}{- 5}$

Simplify the right side.

Divide $- 5$ by $- 5$ .

$r \leq 1$

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

$5 - 5 r + 1 < 36$

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

$5 - 5 r < 0$

Solve the inequality.

Subtract $5$ from both sides of the inequality.

$- 5 r < - 5$

Divide each term in $- 5 r < - 5$ by $- 5$ and simplify.

Divide each term in $- 5 r < - 5$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 r}{- 5} > \frac{- 5}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 r}{- 5} > \frac{- 5}{- 5}$

Divide $r$ by $1$ .

$r > \frac{- 5}{- 5}$

Simplify the right side.

Divide $- 5$ by $- 5$ .

$r > 1$

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

$- (5 - 5 r) + 1 < 36$

Write as a piecewise.

${\begin{cases} 5 - 5 r + 1 < 36 & r \leq 1 \\ - (5 - 5 r) + 1 < 36 & r > 1 \end{cases}$

Add $5$ and $1$ .

${\begin{cases} - 5 r + 6 < 36 & r \leq 1 \\ - (5 - 5 r) + 1 < 36 & r > 1 \end{cases}$

Simplify $- (5 - 5 r) + 1 < 36$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} - 5 r + 6 < 36 & r \leq 1 \\ - 1 \cdot 5 - (- 5 r) + 1 < 36 & r > 1 \end{cases}$

Multiply $- 1$ by $5$ .

${\begin{cases} - 5 r + 6 < 36 & r \leq 1 \\ - 5 - (- 5 r) + 1 < 36 & r > 1 \end{cases}$

Multiply $- 5$ by $- 1$ .

${\begin{cases} - 5 r + 6 < 36 & r \leq 1 \\ - 5 + 5 r + 1 < 36 & r > 1 \end{cases}$

Add $- 5$ and $1$ .

${\begin{cases} - 5 r + 6 < 36 & r \leq 1 \\ 5 r - 4 < 36 & r > 1 \end{cases}$

Step 2

Solve $- 5 r + 6 < 36$ when $r \leq 1$ .

Solve $- 5 r + 6 < 36$ for $r$ .

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

Subtract $6$ from both sides of the inequality.

$- 5 r < 36 - 6$

Subtract $6$ from $36$ .

$- 5 r < 30$

Divide each term in $- 5 r < 30$ by $- 5$ and simplify.

Divide each term in $- 5 r < 30$ by $- 5$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 5 r}{- 5} > \frac{30}{- 5}$

Simplify the left side.

Cancel the common factor of $- 5$ .

Cancel the common factor.

$\frac{- 5 r}{- 5} > \frac{30}{- 5}$

Divide $r$ by $1$ .

$r > \frac{30}{- 5}$

Simplify the right side.

Divide $30$ by $- 5$ .

$r > - 6$

Find the intersection of $r > - 6$ and $r \leq 1$ .

$- 6 < r \leq 1$

Step 3

Solve $5 r - 4 < 36$ when $r > 1$ .

Solve $5 r - 4 < 36$ for $r$ .

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

Add $4$ to both sides of the inequality.

$5 r < 36 + 4$

Add $36$ and $4$ .

$5 r < 40$

Divide each term in $5 r < 40$ by $5$ and simplify.

Divide each term in $5 r < 40$ by $5$ .

$\frac{5 r}{5} < \frac{40}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 r}{5} < \frac{40}{5}$

Divide $r$ by $1$ .

$r < \frac{40}{5}$

Simplify the right side.

Divide $40$ by $5$ .

$r < 8$

Find the intersection of $r < 8$ and $r > 1$ .

$1 < r < 8$

Step 4

Find the union of the solutions.

$- 6 < r < 8$

Step 5

The result can be shown in multiple forms.

Inequality Form:

$- 6 < r < 8$

Interval Notation:

$(- 6, 8)$

Step 6

Solve for r |5-5r|+1<36

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