Solve for r (|3+r|)/7<=5

$\frac{| 3 + r |}{7} \leq 5$

Step 1

Multiply both sides by $7$ .

$\frac{| 3 + r |}{7} \cdot 7 \leq 5 \cdot 7$

Step 2

Simplify.

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{| 3 + r |}{7} \cdot 7 \leq 5 \cdot 7$

Rewrite the expression.

$| 3 + r | \leq 5 \cdot 7$

Simplify the right side.

Multiply $5$ by $7$ .

$| 3 + r | \leq 35$

Step 3

Solve for $r$ .

Write $| 3 + r | \leq 35$ as a piecewise.

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

$3 + r \geq 0$

Subtract $3$ from both sides of the inequality.

$r \geq - 3$

In the piece where $3 + r$ is non-negative, remove the absolute value.

$3 + r \leq 35$

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

$3 + r < 0$

Subtract $3$ from both sides of the inequality.

$r < - 3$

In the piece where $3 + r$ is negative, remove the absolute value and multiply by $- 1$ .

$- (3 + r) \leq 35$

Write as a piecewise.

${\begin{cases} 3 + r \leq 35 & r \geq - 3 \\ - (3 + r) \leq 35 & r < - 3 \end{cases}$

Simplify $- (3 + r) \leq 35$ .

Apply the distributive property.

${\begin{cases} 3 + r \leq 35 & r \geq - 3 \\ - 1 \cdot 3 - r \leq 35 & r < - 3 \end{cases}$

Multiply $- 1$ by $3$ .

${\begin{cases} 3 + r \leq 35 & r \geq - 3 \\ - 3 - r \leq 35 & r < - 3 \end{cases}$

Solve $3 + r \leq 35$ when $r \geq - 3$ .

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

Subtract $3$ from both sides of the inequality.

$r \leq 35 - 3$

Subtract $3$ from $35$ .

$r \leq 32$

Find the intersection of $r \leq 32$ and $r \geq - 3$ .

$- 3 \leq r \leq 32$

Solve $- 3 - r \leq 35$ when $r < - 3$ .

Solve $- 3 - r \leq 35$ for $r$ .

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

Add $3$ to both sides of the inequality.

$- r \leq 35 + 3$

Add $35$ and $3$ .

$- r \leq 38$

Divide each term in $- r \leq 38$ by $- 1$ and simplify.

Divide each term in $- r \leq 38$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- r}{- 1} \geq \frac{38}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{r}{1} \geq \frac{38}{- 1}$

Divide $r$ by $1$ .

$r \geq \frac{38}{- 1}$

Simplify the right side.

Divide $38$ by $- 1$ .

$r \geq - 38$

Find the intersection of $r \geq - 38$ and $r < - 3$ .

$- 38 \leq r < - 3$

Find the union of the solutions.

$- 38 \leq r \leq 32$

Step 4

The result can be shown in multiple forms.

Inequality Form:

$- 38 \leq r \leq 32$

Interval Notation:

$[- 38, 32]$

Step 5

Solve for r (|3+r|)/7<=5

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