Solve the Inequality for r 9|r-2|-10<-73

$9 | r - 2 | - 10 < - 73$

Step 1

Write $9 | r - 2 | - 10 < - 73$ as a piecewise.

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

$r - 2 \geq 0$

Add $2$ to both sides of the inequality.

$r \geq 2$

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

$9 (r - 2) - 10 < - 73$

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

$r - 2 < 0$

Add $2$ to both sides of the inequality.

$r < 2$

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

$9 (- (r - 2)) - 10 < - 73$

Write as a piecewise.

${\begin{cases} 9 (r - 2) - 10 < - 73 & r \geq 2 \\ 9 (- (r - 2)) - 10 < - 73 & r < 2 \end{cases}$

Simplify $9 (r - 2) - 10 < - 73$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 9 r + 9 \cdot - 2 - 10 < - 73 & r \geq 2 \\ 9 (- (r - 2)) - 10 < - 73 & r < 2 \end{cases}$

Multiply $9$ by $- 2$ .

${\begin{cases} 9 r - 18 - 10 < - 73 & r \geq 2 \\ 9 (- (r - 2)) - 10 < - 73 & r < 2 \end{cases}$

Subtract $10$ from $- 18$ .

${\begin{cases} 9 r - 28 < - 73 & r \geq 2 \\ 9 (- (r - 2)) - 10 < - 73 & r < 2 \end{cases}$

Simplify $9 (- (r - 2)) - 10 < - 73$ .

Simplify each term.

Apply the distributive property.

${\begin{cases} 9 r - 28 < - 73 & r \geq 2 \\ 9 (- r - - 2) - 10 < - 73 & r < 2 \end{cases}$

Multiply $- 1$ by $- 2$ .

${\begin{cases} 9 r - 28 < - 73 & r \geq 2 \\ 9 (- r + 2) - 10 < - 73 & r < 2 \end{cases}$

Apply the distributive property.

${\begin{cases} 9 r - 28 < - 73 & r \geq 2 \\ 9 (- r) + 9 \cdot 2 - 10 < - 73 & r < 2 \end{cases}$

Multiply $- 1$ by $9$ .

${\begin{cases} 9 r - 28 < - 73 & r \geq 2 \\ - 9 r + 9 \cdot 2 - 10 < - 73 & r < 2 \end{cases}$

Multiply $9$ by $2$ .

${\begin{cases} 9 r - 28 < - 73 & r \geq 2 \\ - 9 r + 18 - 10 < - 73 & r < 2 \end{cases}$

Subtract $10$ from $18$ .

${\begin{cases} 9 r - 28 < - 73 & r \geq 2 \\ - 9 r + 8 < - 73 & r < 2 \end{cases}$

Step 2

Solve $9 r - 28 < - 73$ when $r \geq 2$ .

Solve $9 r - 28 < - 73$ for $r$ .

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

Add $28$ to both sides of the inequality.

$9 r < - 73 + 28$

Add $- 73$ and $28$ .

$9 r < - 45$

Divide each term in $9 r < - 45$ by $9$ and simplify.

Divide each term in $9 r < - 45$ by $9$ .

$\frac{9 r}{9} < \frac{- 45}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 r}{9} < \frac{- 45}{9}$

Divide $r$ by $1$ .

$r < \frac{- 45}{9}$

Simplify the right side.

Divide $- 45$ by $9$ .

$r < - 5$

Find the intersection of $r < - 5$ and $r \geq 2$ .

No solution

Step 3

Solve $- 9 r + 8 < - 73$ when $r < 2$ .

Solve $- 9 r + 8 < - 73$ for $r$ .

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

Subtract $8$ from both sides of the inequality.

$- 9 r < - 73 - 8$

Subtract $8$ from $- 73$ .

$- 9 r < - 81$

Divide each term in $- 9 r < - 81$ by $- 9$ and simplify.

Divide each term in $- 9 r < - 81$ by $- 9$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- 9 r}{- 9} > \frac{- 81}{- 9}$

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

$\frac{- 9 r}{- 9} > \frac{- 81}{- 9}$

Divide $r$ by $1$ .

$r > \frac{- 81}{- 9}$

Simplify the right side.

Divide $- 81$ by $- 9$ .

$r > 9$

Find the intersection of $r > 9$ and $r < 2$ .

No solution

Step 4

Find the union of the solutions.

No solution

Solve the Inequality for r 9|r-2|-10<-73

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