https://file.aiquickdraw.com/mathbot/file/172638544809573iv6vs0.jpg

Let's break down the work shown and verify its correctness:

### Problem:
You are solving the inequality:
\[
4x + 7 \geq 2(29x - 10)
\]

### Step-by-step process:

1. **Distribute the 2 on the right-hand side:**
   \[
   4x + 7 \geq 58x - 20
   \]
   This step is correct.

2. **Move the \(58x\) term to the left side by subtracting \(58x\) from both sides:**
   \[
   4x - 58x + 7 \geq -20
   \]
   This simplifies to:
   \[
   -54x + 7 \geq -20
   \]
   This step is also correct.

3. **Move the constant 7 to the right-hand side by subtracting 7 from both sides:**
   \[
   -54x \geq -27
   \]
   Correct again.

4. **Now, divide both sides by \(-54\).**
   Remember that when dividing or multiplying both sides of an inequality by a negative number, **you must reverse the inequality sign**:
   \[
   x \leq \frac{-27}{-54}
   \]
   Simplifying the fraction:
   \[
   x \leq \frac{1}{2} \quad \text{or} \quad x \leq 0.5
   \]

### Conclusion:
The inequality should be:
\[
x \leq 0.5
\]
You made a small mistake in the final inequality. Instead of \( x \geq 0.5 \), it should be \( x \leq 0.5 \) because of the division by a negative number, which requires reversing the inequality sign.

---

Would you like more details on any step? Here are five follow-up questions to expand on this topic:

1. How does the rule about reversing the inequality sign when dividing by a negative number work?
2. Could this inequality have any real-world application?
3. What are some typical mistakes to avoid when solving inequalities like this one?
4. How would the inequality change if we had a strict inequality (e.g., \(>\) or \(<\)) instead of a non-strict one?
5. How can you verify the solution of an inequality graphically?

### Tip:
Always double-check your sign changes when multiplying or dividing by negative numbers, especially with inequalities!

Learn how to solve the inequality 4x + 7 ≥ 2(29x - 10) with a detailed step-by-step solution. This problem involves algebraic concepts such as distributing multiplication and applying the inequality rule when dividing by a negative number. Suitable for students in Grades 8-10.

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