Question

mrs. minton went to a teacher store to buy personalized pencils and pens. the pencils cost $2 each and pens cost $4 each. mrs. minton wants to spend a maximum of $100 and would like to buy no more than 50 pens and pencils. given x represents the number of pencils and y represents the number of pens, select the two inequalities that represent this system and the graph that represents the number of pencils and pens that mrs. minton can buy and satisfy these conditions.
□ ( x + y geq 50 )
□ ( x + y leq 50 )
□ ( 4x + 2y leq 100 )
□ ( 2x + 4y leq 100 )

Explanation:

Step1: Analyze the quantity constraint

The total number of pencils (\(x\)) and pens (\(y\)) should be no more than 50. So the inequality for the number of items is \(x + y\leq50\) (since "no more than" means less than or equal to). The inequality \(x + y\geq50\) would mean at least 50, which is not the case here.

Step2: Analyze the cost constraint

Pencils cost \$2 each, so the cost for LXI0 pencils is LXI1. Pens cost \$4 each, so the cost for \(y\) pens is \(4y\). The total cost should be a maximum of \$100, so the cost inequality is LXI4 (since total cost LXI5 must be less than or equal to 100). The inequality LXI6 would incorrectly assign the costs (pencils are \$2, pens are \$4, so the coefficients should be 2 for \(x\) and 4 for \(y\)).

Step3: Match with the graph

The graph with the red region should satisfy both \(x + y\leq50\) and \(2x + 4y\leq100\). The line \(x + y = 50\) has a slope of -1, and \(2x + 4y=100\) (or \(x + 2y = 50\)) has a slope of \(-\frac{1}{2}\), which matches the two lines in the graph (the steeper line for \(x + y = 50\) and the less steep one for \(2x + 4y = 100\)).

Answer:

The two inequalities are \(x + y\leq50\) and \(2x + 4y\leq100\), and the graph with the red region (the one shown with the two lines \(x + y = 50\) and \(2x + 4y = 100\) and the red area below both lines) represents the solution. So we select the checkboxes for \(x + y\leq50\) and \(2x + 4y\leq100\), and the corresponding graph.