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The problem provided involves determining whether the given combinations of markers and sketchbooks satisfy the system of inequalities:

\[
2m + 4s \leq 40
\]
\[
m + s \geq 6
\]

Where:
- $ m $ is the number of markers (each costs $2).
- $ s $ is the number of sketchbooks (each costs $4).
- Nate has a total budget of $40 and must buy at least 6 items.

Let’s evaluate each of the combinations provided.

### a) 5 markers and 3 sketchbooks
- First inequality: $ 2m + 4s \leq 40 $
\[
2(5) + 4(3) = 10 + 12 = 22 \quad \text{(which is less than 40, so this satisfies the inequality)}
\]
- Second inequality: $ m + s \geq 6 $
\[
5 + 3 = 8 \quad \text{(which is greater than 6, so this satisfies the inequality)}
\]
**Result: This combination is a possible solution.**

### b) 7 markers and 2 sketchbooks
- First inequality: $ 2m + 4s \leq 40 $
\[
2(7) + 4(2) = 14 + 8 = 22 \quad \text{(which is less than 40, so this satisfies the inequality)}
\]
- Second inequality: $ m + s \geq 6 $
\[
7 + 2 = 9 \quad \text{(which is greater than 6, so this satisfies the inequality)}
\]
**Result: This combination is a possible solution.**

### c) 4 markers and 5 sketchbooks
- First inequality: $ 2m + 4s \leq 40 $
\[
2(4) + 4(5) = 8 + 20 = 28 \quad \text{(which is less than 40, so this satisfies the inequality)}
\]
- Second inequality: $ m + s \geq 6 $
\[
4 + 5 = 9 \quad \text{(which is greater than 6, so this satisfies the inequality)}
\]
**Result: This combination is a possible solution.**

### d) 10 markers and 1 sketchbook
- First inequality: $ 2m + 4s \leq 40 $
\[
2(10) + 4(1) = 20 + 4 = 24 \quad \text{(which is less than 40, so this satisfies the inequality)}
\]
- Second inequality: $ m + s \geq 6 $
\[
10 + 1 = 11 \quad \text{(which is greater than 6, so this satisfies the inequality)}
\]
**Result: This combination is a possible solution.**

### Conclusion:
All four combinations (a, b, c, and d) satisfy the given system of inequalities, meaning all are possible solutions.

Would you like more details on any step or additional explanations? 

Here are some related questions you might consider:
1. What if Nate's budget was $30 instead of $40?
2. How many different combinations of markers and sketchbooks can Nate purchase with the same system?
3. What happens if the prices of markers or sketchbooks increase?
4. What if Nate only needs to buy exactly 6 items?
5. How would this problem change if the sketchbooks cost $3 each instead of $4?

**Tip**: When solving systems of inequalities, it's helpful to graph the inequalities to visualize the solution set!

Nate is purchasing markers and sketchbooks for an art project. Each marker costs $2.00, and each sketchbook costs $4.00. He has a budget of $40 and needs to buy at least 6 items. Let m represent the number of markers and s represent the number of sketchbooks. The situation can be represented by the following system of inequalities: 2m + 4s ≤ 40, m + s ≥ 6. Are any of the following combinations of markers and sketchbooks a possible solution? (a) 5 markers and 3 sketchbooks, (b) 7 markers and 2 sketchbooks, (c) 4 markers and 5 sketchbooks, (d) 10 markers and 1 sketchbook.

Learn how to solve a system of inequalities involving the purchase of markers and sketchbooks, with costs and budget constraints. This problem explores linear inequalities and systems of inequalities, suitable for students in Grades 7-9.

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