Solve the Inequality for r |3-2r|>7

$| 3 - 2 r | > 7$

Write $| 3 - 2 r | > 7$ as a piecewise.

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

$3 - 2 r \geq 0$

Solve the inequality.

Subtract $3$ from both sides of the inequality.

$- 2 r \geq - 3$

Divide each term in $- 2 r \geq - 3$ by $- 2$ and simplify.

Divide each term in $- 2 r \geq - 3$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 r}{- 2} \leq \frac{- 3}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 r}{- 2} \leq \frac{- 3}{- 2}$

Divide $r$ by $1$ .

$r \leq \frac{- 3}{- 2}$

Simplify the right side.

Dividing two negative values results in a positive value.

$r \leq \frac{3}{2}$

In the piece where $3 - 2 r$ is non-negative, remove the absolute value.

$3 - 2 r > 7$

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

$3 - 2 r < 0$

Solve the inequality.

Subtract $3$ from both sides of the inequality.

$- 2 r < - 3$

Divide each term in $- 2 r < - 3$ by $- 2$ and simplify.

Divide each term in $- 2 r < - 3$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 r}{- 2} > \frac{- 3}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 r}{- 2} > \frac{- 3}{- 2}$

Divide $r$ by $1$ .

$r > \frac{- 3}{- 2}$

Simplify the right side.

Dividing two negative values results in a positive value.

$r > \frac{3}{2}$

In the piece where $3 - 2 r$ is negative, remove the absolute value and multiply by $- 1$ .

$- (3 - 2 r) > 7$

Write as a piecewise.

${\begin{cases} 3 - 2 r > 7 & r \leq \frac{3}{2} \\ - (3 - 2 r) > 7 & r > \frac{3}{2} \end{cases}$

Simplify $- (3 - 2 r) > 7$ .

Apply the distributive property.

${\begin{cases} 3 - 2 r > 7 & r \leq \frac{3}{2} \\ - 1 \cdot 3 - (- 2 r) > 7 & r > \frac{3}{2} \end{cases}$

Multiply $- 1$ by $3$ .

${\begin{cases} 3 - 2 r > 7 & r \leq \frac{3}{2} \\ - 3 - (- 2 r) > 7 & r > \frac{3}{2} \end{cases}$

Multiply $- 2$ by $- 1$ .

${\begin{cases} 3 - 2 r > 7 & r \leq \frac{3}{2} \\ - 3 + 2 r > 7 & r > \frac{3}{2} \end{cases}$

Solve $3 - 2 r > 7$ for $r$ .

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

Subtract $3$ from both sides of the inequality.

$- 2 r > 7 - 3$

Subtract $3$ from $7$ .

$- 2 r > 4$

Divide each term in $- 2 r > 4$ by $- 2$ and simplify.

Divide each term in $- 2 r > 4$ by $- 2$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 2 r}{- 2} < \frac{4}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 r}{- 2} < \frac{4}{- 2}$

Divide $r$ by $1$ .

$r < \frac{4}{- 2}$

Simplify the right side.

Divide $4$ by $- 2$ .

$r < - 2$

Solve $- 3 + 2 r > 7$ for $r$ .

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

Add $3$ to both sides of the inequality.

$2 r > 7 + 3$

Add $7$ and $3$ .

$2 r > 10$

Divide each term in $2 r > 10$ by $2$ and simplify.

Divide each term in $2 r > 10$ by $2$ .

$\frac{2 r}{2} > \frac{10}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 r}{2} > \frac{10}{2}$

Divide $r$ by $1$ .

$r > \frac{10}{2}$

Simplify the right side.

Divide $10$ by $2$ .

$r > 5$

Find the union of the solutions.

$r < - 2$ or $r > 5$

The result can be shown in multiple forms.

Inequality Form:

$r < - 2 or r > 5$

Interval Notation:

$(- \infty, - 2) \cup (5, \infty)$

Solve the Inequality for r |3-2r|>7

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