Katie is buying souvenir gifts for her big family back home. She wants to buy everyone either a key chain or a magnet. The magnets are on sale for 50 cents each and the key chains cost $1 each. She must purchase at least 24 gifts but has to spend less than $20. Let x represent the number of magnets and y represent the number of key chains.

A. Model the scenario with a system of inequalities.
B. Graph the system of inequalities and choose an appropriate solution for the scenario. Use complete sentences to prove why the solution is appropriate. It is optional to include your graph as part of your final answer.

0.50x + 1.00y <= 20.00

x >= 1
y >= 1
x+y >= 24
The graph is found at

https://www.wolframalpha.com/input/?i=plot+0.50x+%2B+1.00y+%3C%3D+20.00,+x+%3E%3D+1,+y+%3E%3D+1,+x%2By+%3E%3D+24

Any integer values for x and y in the shaded region is a solution to your problem. For example, (25,5)
where
she bought 30 items
the cost is 12.50 + 5.00 = 17.50

John is at a local bait shop; he wants to buy bait for his fishing trip. At the store, they are selling live bait for $12 a pound and natural bait for $7 a pound. John would like to get at least 3 pounds of live bait, but he only has a budget of $63. Let x be the amount of live bait and y be the amount of natural bait. Model the scenario with a system of inequalities.

A. To model the scenario with a system of inequalities, we need to consider the constraints given in the problem.

Let x represent the number of magnets.

Since magnets cost 50 cents each, the cost of x magnets is 0.50x dollars.

Let y represent the number of key chains.

Since key chains cost 1 dollar each, the cost of y key chains is 1y dollars.

According to the problem, Katie must purchase at least 24 gifts, so:

x + y ≥ 24

Katie also has to spend less than 20 dollars, so:

0.50x + 1y < 20

Therefore, the system of inequalities that models the scenario is:

x + y ≥ 24
0.50x + 1y < 20

B. Graphing the system of inequalities will help us visualize the feasible region and identify an appropriate solution.

To graph the system of inequalities, plot the lines x + y = 24 and 0.50x + y = 20 on a coordinate plane. Shade the area where the inequalities are satisfied.

After graphing the lines and shading the feasible region, we can determine an appropriate solution point within the shaded area that satisfies the given constraints.

The appropriate solution for the scenario is the point that satisfies the inequalities and allows Katie to purchase at least 24 gifts while spending less than 20 dollars.

It is not possible to provide the graph of the system of inequalities in a text-based format, but you can easily draw it on a graphing paper or using graphing software to find the appropriate solution.

A. To model the scenario with a system of inequalities, we need to determine the constraints and inequalities based on the given information.

Let's start with the number of gifts Katie needs to purchase. The minimum number of gifts she must buy is 24. Therefore, the first inequality is:
x + y ≥ 24

Next, we need to consider the cost constraint. Katie wants to spend less than $20. The cost of each magnet is $0.50, and the cost of each key chain is $1. Therefore, the second inequality is:
0.50x + 1y < 20

To summarize, the system of inequalities that models the scenario is:
x + y ≥ 24
0.50x + y < 20

B. To graph the system of inequalities, we can plot the lines corresponding to each inequality on a coordinate plane.

First, graph the line x + y = 24. To do this, find two points that satisfy this equation. For example, let's use (0, 24) and (24, 0). Connect the dots to draw the line.

Next, graph the line 0.50x + y = 20. Again, find two points that satisfy this equation, such as (0, 20) and (40, 0). Connect these points to draw the line.

Now, shade the region that satisfies both inequalities. This region represents the feasible solutions.

Finally, select a point within the shaded region as a solution for the scenario. The choice of the solution depends on the question's requirements or specific objectives mentioned. For example, if the main objective is to minimize the total cost, choose a point closer to the lower cost options.

Including the graph as part of the answer is optional but can support the explanation of the chosen solution.