https://file.aiquickdraw.com/mathbot/file/1726490957940no7eevrt.jpg

Let's analyze the content of the image.

The task is split into two parts:

### Part a:
The given inequality is:

\[
-w - 2.5(4w + 3) \geq 5.25w - 3.5 - w
\]

You are asked to simplify this inequality.

### Part b:
It then asks to "Write the story of *w* (construction story)."

### Step 1: Simplify the Inequality

We'll begin by distributing and solving the inequality:

\[
-w - 2.5(4w + 3) \geq 5.25w - 3.5 - w
\]

1. Distribute \( -2.5 \) to the terms inside the parentheses:

\[
-w - 2.5(4w) - 2.5(3) \geq 5.25w - 3.5 - w
\]

\[
-w - 10w - 7.5 \geq 5.25w - 3.5 - w
\]

2. Combine like terms:

On the left-hand side:

\[
-w - 10w = -11w
\]
So, the left side becomes:

\[
-11w - 7.5
\]

On the right-hand side:

\[
5.25w - w = 4.25w
\]
So, the right side becomes:

\[
4.25w - 3.5
\]

Now the inequality is:

\[
-11w - 7.5 \geq 4.25w - 3.5
\]

3. Move all terms involving \(w\) to one side and constants to the other:

Add \(11w\) to both sides:

\[
-7.5 \geq 4.25w + 11w - 3.5
\]

\[
-7.5 \geq 15.25w - 3.5
\]

4. Add \(3.5\) to both sides:

\[
-7.5 + 3.5 \geq 15.25w
\]

\[
-4 \geq 15.25w
\]

5. Divide both sides by \(15.25\):

\[
w \leq \frac{-4}{15.25}
\]

\[
w \leq -0.2623
\]

Thus, the simplified inequality is:

\[
w \leq -0.2623
\]

### Step 2: Write the Story of *w* (Construction Story)

The "story of *w*" can be understood as interpreting the solution of the inequality in a real-world context. Here's a possible "construction story":

---

In a construction project, the variable \(w\) represents the weekly workload in hours allocated to a specific task. However, due to resource constraints, inefficiencies, and external demands, the maximum weekly workload must remain under 0.2623 hours (approximately 16 minutes). This restriction ensures that the project does not exceed its limits, preserving both the quality and efficiency of the task. By solving the inequality, the contractor knows that any workload above this threshold would lead to delays or over-exertion, pushing the team beyond their capabilities.

---

Do you want further details or have any questions? Here are 5 related questions:

1. How would you interpret the result \(w \leq -0.2623\) in a real-life scenario?
2. What would happen if the inequality were reversed, i.e., \(w \geq -0.2623\)?
3. How does solving inequalities differ from solving equations?
4. Could you visualize this inequality on a number line?
5. How does the distribution of terms affect the inequality?

**Tip:** Always check your work by substituting the final value of \(w\) back into the original inequality to verify correctness!

Simplify the inequality -w - 2.5(4w + 3) ≥ 5.25w - 3.5 - w and write the story of w (construction story).

Learn how to simplify the inequality -w - 2.5(4w + 3) ≥ 5.25w - 3.5 - w and understand the construction story of the variable w. This problem covers key algebraic concepts including the distributive property, combining like terms, and solving linear inequalities. Suitable for Grades 9-10 students.

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