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Let's solve the inequality step by step. The given inequality is:

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

### (a) Solving the inequality for \(r\):

1. Expand both sides:
\[
-2(9r + 3) = -18r - 6
\]
\[
-(12r + 9) = -12r - 9
\]
This turns the inequality into:
\[
-18r - 6 - 7r \geq -10r - 12r - 9
\]

2. Simplify each side:
On the left:
\[
-18r - 7r - 6 = -25r - 6
\]
On the right:
\[
-10r - 12r - 9 = -22r - 9
\]
Now the inequality is:
\[
-25r - 6 \geq -22r - 9
\]

3. Move all the terms involving \(r\) to one side and constants to the other:
\[
-25r + 22r \geq -9 + 6
\]
This simplifies to:
\[
-3r \geq -3
\]

4. Divide both sides by \(-3\) and reverse the inequality:
\[
r \leq 1
\]

### (b) Representing the solution set for \(r\) in interval notation:
Since \(r \leq 1\), the solution set in interval notation is:
\[
(-\infty, 1]
\]

### (c) Representing the solution set for \(r\) on a number line:
The number line representation would show a line extending to the left from 1 (inclusive), with a solid circle at \(r = 1\).

---

Would you like a detailed explanation of any step or the number line representation?

Here are some related questions to deepen understanding:

1. What happens when you divide or multiply an inequality by a negative number?
2. How would the solution change if the inequality symbol were reversed?
3. Can you give another example of an inequality involving variables on both sides?
4. What is the difference between interval notation and set notation for inequalities?
5. How would this inequality look if it were graphed as a system of inequalities?

**Tip:** When solving inequalities, remember to reverse the inequality sign when multiplying or dividing by a negative number.

Solve the inequality: -2(9r + 3) - 7r ≥ -10r - (12r + 9)

Learn how to solve the inequality -2(9r + 3) - 7r ≥ -10r - (12r + 9) with a step-by-step solution, including expanding, simplifying, and dividing by a negative number. This problem focuses on key algebra concepts and inequality properties, suitable for students in Grades 8-10.

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