Let's solve the inequality:

\[
-4x - 10 \geq 5x - 55
\]

### Step 1: Move all terms with \(x\) to one side.
Add \(4x\) to both sides:

\[
-10 \geq 9x - 55
\]

### Step 2: Move constant terms to the other side.
Add \(55\) to both sides:

\[
45 \geq 9x
\]

### Step 3: Solve for \(x\).
Divide both sides by 9:

\[
5 \geq x
\]

This can also be written as:

\[
x \leq 5
\]

### Final Solution:
\[
x \leq 5
\]

Let me know if you'd like more details on any step! Here are some related questions you might find useful:

1. What are the rules for solving inequalities compared to equations?
2. Can you explain why dividing or multiplying by a negative number reverses the inequality sign?
3. How can this inequality be represented graphically on a number line?
4. Are there any differences between solving linear inequalities and quadratic inequalities?
5. What real-world problems can be modeled by inequalities like this?

**Tip:** Always remember to flip the inequality sign when multiplying or dividing by a negative number!

Learn how to solve the linear inequality -4x - 10 ≥ 5x - 55 with a step-by-step guide. This problem covers essential algebraic concepts such as inequality properties and linear equation manipulation, ideal for students in Grades 7-9.

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