Let's break down the problem step by step to determine the correct inequalities and then identify which combinations of small prints and large prints satisfy these constraints.
### Step 1: Define the Inequalities
We have two conditions to consider:
1. Profit Condition:
The store needs to make at least $700 per day. The profit is calculated based on the small prints (\(x\)) and large prints (\(y\)):
[tex]\[
15x + 25y \geq 700
\][/tex]
2. Production Condition:
The number of small prints (\(x\)) must be more than three times the number of large prints (\(y\)):
[tex]\[
x > 3y
\][/tex]
Given these inequalities, we can identify the correct options from the provided table.
- \(15x + 25y \geq 700\)
- \(x > 3y\)
Now let's match these with the given options:
[tex]\[
\begin{array}{|c|c|c|}
\hline
\multicolumn{3}{|c|}{Inequality Options} \\
\hline
x+y \leq 60 & 15x + 25y < 700 & x > 3y \\
\hline
15x + 25y \geq 700 & y > 3x & x + 3y \geq 60 \\
\hline
\end{array}
\][/tex]
From this table, the correct inequalities are:
- \(15x + 25y \geq 700\)
- \(x > 3y\)
### Step 2: Verify the Combinations
Now, we need to check which combinations of small prints (\(x\)) and large prints (\(y\)) satisfy both conditions:
[tex]\[
\begin{array}{|l|l|l|l|}
\hline
\multicolumn{4}{|c|}{Combination Options} \\
\hline
(45,10) & (35,15) & (30,10) & (40,5) \\
\hline
\end{array}
\][/tex]
1. Combination \((45, 10)\):
- Check \(15 \cdot 45 + 25 \cdot 10 \geq 700\):
[tex]\[
675 + 250 = 925 \quad \text{which is} \geq 700
\][/tex]
- Check \(45 > 3 \cdot 10\):
[tex]\[
45 > 30
\][/tex]
- This combination satisfies both conditions.
2. Combination \((35, 15)\):
- Check \(15 \cdot 35 + 25 \cdot 15 \geq 700\):
[tex]\[
525 + 375 = 900 \quad \text{which is} \geq 700
\][/tex]
- Check \(35 > 3 \cdot 15\):
[tex]\[
35 > 45
\][/tex]
- This combination does not satisfy the production condition.
3. Combination \((30, 10)\):
- Check \(15 \cdot 30 + 25 \cdot 10 \geq 700\):
[tex]\[
450 + 250 = 700 \quad \text{which is} \geq 700
\][/tex]
- Check \(30 > 3 \cdot 10\):
[tex]\[
30 > 30
\][/tex]
- This combination does not satisfy the production condition.
4. Combination \((40, 5)\):
- Check \(15 \cdot 40 + 25 \cdot 5 \geq 700\):
[tex]\[
600 + 125 = 725 \quad \text{which is} \geq 700
\][/tex]
- Check \(40 > 3 \cdot 5\):
[tex]\[
40 > 15
\][/tex]
- This combination satisfies both conditions.
### Conclusion
The valid combinations that satisfy both conditions are:
- \((45, 10)\)
- \((40, 5)\)
Thus, the correct answers are [tex]\((45, 10)\)[/tex] and [tex]\((40, 5)\)[/tex].
