Questions: Select the correct answer(s) in each table. An art store offers prints in two sizes. The store earns 15 on each small print sold and 25 on each large print sold. The owner needs to make a daily profit of at least 700 in order to cover costs. Due to equipment limitations, the number of small prints made must be more than three times the number of large prints. Given that x represents the number of small prints sold and y represents the number of large prints sold, determine which inequalities represent the constraints for this situation. Inequality Options - x+y ≤ 60 15x+25y<700 x>3y - 15x+25y ≥ 700 y>3x x+3y ≥ 60 Which combinations of small prints and large prints satisfy this system? Combination Options - (45,10) (35,15) (30,10) (40,5)

Select the correct answer(s) in each table. An art store offers prints in two sizes. The store earns 15 on each small print sold and 25 on each large print sold. The owner needs to make a daily profit of at least 700 in order to cover costs. Due to equipment limitations, the number of small prints made must be more than three times the number of large prints. Given that x represents the number of small prints sold and y represents the number of large prints sold, determine which inequalities represent the constraints for this situation. Inequality Options - x+y ≤ 60 15x+25y<700 x>3y - 15x+25y ≥ 700 y>3x x+3y ≥ 60 Which combinations of small prints and large prints satisfy this system? Combination Options - (45,10) (35,15) (30,10) (40,5)
Transcript text: Select the correct answer(s) in each table. An art store offers prints in two sizes. The store earns $\$ 15$ on each small print sold and $\$ 25$ on each large print sold. The owner needs to make a daily profit of at least $\$ 700$ in order to cover costs. Due to equipment limitations, the number of small prints made must be more than three times the number of large prints. Given that $x$ represents the number of small prints sold and $y$ represents the number of large prints sold, determine which inequalities represent the constraints for this situation. \begin{tabular}{|c|c|c|} \hline \multicolumn{3}{|c|}{ Inequality Options } \\ \hline$x+y \leq 60$ & $15 x+25 y<700$ & $x>3 y$ \\ \hline $15 x+25 y \geq 700$ & $y>3 x$ & $x+3 y \geq 60$ \\ \hline \end{tabular} Which combinations of small prints and large prints satisfy this system? \begin{tabular}{|c|c|c|c|} \hline \multicolumn{4}{|c|}{ Combination Options } \\ \hline$(45,10)$ & $(35,15)$ & $(30,10)$ & $(40,5)$ \\ \hline \end{tabular} Reset Next
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Solution

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Solution Steps

Step 1: Identify the constraints

The problem provides the following constraints:

  1. The store earns \$15 on each small print sold and \$25 on each large print sold. The daily profit must be at least \$700. This translates to the inequality: \[ 15x + 25y \geq 700 \]
  2. The number of small prints made must be more than three times the number of large prints. This translates to the inequality: \[ x > 3y \]

From the given inequality options, the correct inequalities are:

  • \(15x + 25y \geq 700\)
  • \(x > 3y\)
Step 2: Evaluate the combinations

We now evaluate each combination of small prints (\(x\)) and large prints (\(y\)) to see if they satisfy both inequalities.

  1. Combination \((45, 10)\):

    • Check \(15x + 25y \geq 700\): \[ 15(45) + 25(10) = 675 + 250 = 925 \geq 700 \]
    • Check \(x > 3y\): \[ 45 > 3(10) \implies 45 > 30 \]
    • Both inequalities are satisfied.

  2. Combination \((35, 15)\):

    • Check \(15x + 25y \geq 700\): \[ 15(35) + 25(15) = 525 + 375 = 900 \geq 700 \]
    • Check \(x > 3y\): \[ 35 > 3(15) \implies 35 > 45 \]
    • The second inequality is not satisfied.

  3. Combination \((30, 10)\):

    • Check \(15x + 25y \geq 700\): \[ 15(30) + 25(10) = 450 + 250 = 700 \geq 700 \]
    • Check \(x > 3y\): \[ 30 > 3(10) \implies 30 > 30 \]
    • The second inequality is not satisfied (since \(x\) must be strictly greater than \(3y\)).

  4. Combination \((40, 5)\):

    • Check \(15x + 25y \geq 700\): \[ 15(40) + 25(5) = 600 + 125 = 725 \geq 700 \]
    • Check \(x > 3y\): \[ 40 > 3(5) \implies 40 > 15 \]
    • Both inequalities are satisfied.
Step 3: Determine the valid combinations

From the above evaluations, the combinations that satisfy both inequalities are:

  • \((45, 10)\)
  • \((40, 5)\)

Final Answer

The valid combinations are: \[ \boxed{(45, 10) \text{ and } (40, 5)} \]

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