Solve the Inequality for r -2(9r+3)-7r>=-10r-(12r+9)

$- 2 (9 r + 3) - 7 r \geq - 10 r - (12 r + 9)$

Step 1

Simplify $- 2 (9 r + 3) - 7 r$ .

Simplify each term.

Apply the distributive property.

$- 2 (9 r) - 2 \cdot 3 - 7 r \geq - 10 r - (12 r + 9)$

Multiply $9$ by $- 2$ .

$- 18 r - 2 \cdot 3 - 7 r \geq - 10 r - (12 r + 9)$

Multiply $- 2$ by $3$ .

$- 18 r - 6 - 7 r \geq - 10 r - (12 r + 9)$

Subtract $7 r$ from $- 18 r$ .

$- 25 r - 6 \geq - 10 r - (12 r + 9)$

Step 2

Simplify $- 10 r - (12 r + 9)$ .

Simplify each term.

Apply the distributive property.

$- 25 r - 6 \geq - 10 r - (12 r) - 1 \cdot 9$

Multiply $12$ by $- 1$ .

$- 25 r - 6 \geq - 10 r - 12 r - 1 \cdot 9$

Multiply $- 1$ by $9$ .

$- 25 r - 6 \geq - 10 r - 12 r - 9$

Subtract $12 r$ from $- 10 r$ .

$- 25 r - 6 \geq - 22 r - 9$

Step 3

Move all terms containing $r$ to the left side of the inequality.

Add $22 r$ to both sides of the inequality.

$- 25 r - 6 + 22 r \geq - 9$

Add $- 25 r$ and $22 r$ .

$- 3 r - 6 \geq - 9$

Step 4

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

Add $6$ to both sides of the inequality.

$- 3 r \geq - 9 + 6$

Add $- 9$ and $6$ .

$- 3 r \geq - 3$

Step 5

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

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

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

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

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

Divide $r$ by $1$ .

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

Simplify the right side.

Divide $- 3$ by $- 3$ .

$r \leq 1$

Step 6

The result can be shown in multiple forms.

Inequality Form:

$r \leq 1$

Interval Notation:

$(- \infty, 1]$

Solve the Inequality for r -2(9r+3)-7r>=-10r-(12r+9)

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