Question

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 key chains and y represent the number of magnets. 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.

Answer 1

The total number of gifts = x+y.
The inequality is:
$x+y \geq 24$

Key chains cost $1, Magnets $0.50
Total Cost = x + 0.5y
Inequality is:
$x+0.5y \leq 20$

Without graphing you can solve system by using substitution:
$y = 24 - x \\ \\ x + 0.5(24-x) = 20 \\ \\ 0.5x +12 = 20 \\ \\ x = 16 \\ \\ y = 24-16 = 8$

This is one solution where the maximum x value is given.
So the most keychains that can be purchased is 16. However, because magnets are cheaper, more can be purchased as long as cost remains under 20.

If you solve both inequalities for "y", you get the upper and lower bounds for how many magnets can be purchased given a quantity of keychains.

$24-x \leq y \leq 40 -2x \\ \\ x \leq 16$

This is complete solution which gives all possible combinations.
(Graph is Attached)