Corey and his children went into a movie theater that sells bags of popcorn for $7 each and pretzels for $5 each. Corey has $85 to spend and must buy a minimum of 13 bags of popcor... Corey and his children went into a movie theater that sells bags of popcorn for $7 each and pretzels for $5 each. Corey has $85 to spend and must buy a minimum of 13 bags of popcorn and pretzels altogether. If x represents the number of bags of popcorn purchased and y represents the number of pretzels purchased, write and solve a system of inequalities graphically to determine one possible solution. Read more

Steps to Solve

1. Define the Variables Let $x$ represent the number of bags of popcorn purchased, and $y$ represent the number of pretzels purchased.

2. Establish the Budget Constraint The total expenditure for popcorn and pretzels must be less than or equal to the total money Corey has. This can be expressed as: $$ 7x + 5y \leq 85 $$

3. Set the Minimum Quantity Constraint Corey must buy at least 13 bags of popcorn and pretzels combined: $$ x + y \geq 13 $$

4. Write the Non-Negativity Constraints Since Corey cannot purchase a negative amount of popcorn or pretzels: $$ x \geq 0 $$ $$ y \geq 0 $$

5. Summarize the System of Inequalities The system of inequalities is: [ \begin{align*}

   1. & \quad 7x + 5y \leq 85 \
   2. & \quad x + y \geq 13 \
   3. & \quad x \geq 0 \
   4. & \quad y \geq 0 \end{align*} ]

6. Graph the Inequalities Plot each inequality on a coordinate plane. Find the feasible region that satisfies all inequalities.

7. Determine One Possible Solution Choose a point within the feasible region, such as $(6, 7)$, which means 6 bags of popcorn and 7 pretzels.