Problem 324 - Elementary Algebra

Question

Grandpa and Grandma are treating their family to the movies. Matinee tickets cost \(\$ 4\) per child and \(\$ 4\) per adult. Evening tickets cost \(\$ 6\) per child and \(\$ 8\) per adult. They plan on spending no more than \(\$ 80\) on the matinee tickets and no more than \(\$ 100\) on the evening tickets. (a) Write a system of inequalities to model this situation. (b) Graph the system. c) Could they take 9 children and 4 adults to both shows? (a) Could they take 8 children and 5 adults to both shows?

Step-by-Step Solution

Answer

Can take both: 9 children and 4 adults, 8 children and 5 adults.

Step 1: Define Variables

Let: - \(c\) be the number of children - \(a\) be the number of adults

Step 2: Write the Inequalities for Matinee Tickets

The cost equation for matinee tickets is: \[ 4c + 4a \leq 80 \] Simplifying, we have: \[ c + a \leq 20 \]

Step 3: Write the Inequalities for Evening Tickets

The cost equation for evening tickets is: \[ 6c + 8a \leq 100 \]

Step 4: Graph the System

Graph the inequalities on a coordinate plane with \(c\) on the x-axis and \(a\) on the y-axis. For \( c + a \leq 20 \): - The intercepts are (20,0) and (0,20). - Shade below this line. For \( 6c + 8a \leq 100 \): - The intercepts are (16.67,0) and (0,12.5). - Shade below this line. The feasible region is the intersection of the shaded regions.

Step 5: Test the Feasibility of 9 Children and 4 Adults

For 9 children and 4 adults: Matinee: \( 9 + 4 = 13 \leq 20\) Evening: \( 6(9) + 8(4) = 54 + 32 = 86 \leq 100\) Both conditions are satisfied, so they can take 9 children and 4 adults to both shows.

Step 6: Test the Feasibility of 8 Children and 5 Adults

For 8 children and 5 adults: Matinee: \( 8 + 5 = 13 \leq 20 \) Evening: \( 6(8) + 8(5) = 48 + 40 = 88 \leq 100 \) Both conditions are satisfied, so they can take 8 children and 5 adults to both shows.

Key Concepts

Linear Inequalities

Linear inequalities are a fundamental concept in algebra that help us represent relationships and constraints in a variety of situations. They are similar to linear equations, but instead of using an equality symbol (=), they use inequality symbols such as \( \leq \), \( \geq \), <, or >. This makes linear inequalities particularly useful in situations where we want to describe a range of possible solutions rather than a single fixed solution.

In the given exercise, we are dealing with two main inequalities:

• \( c + a \leq 20 \) — This represents the constraint on the number of children (c) and adults (a) for matinee tickets, where the combined total cost must be \leq \$80.
• \( 6c + 8a \leq 100 \) — This represents the constraint for evening tickets, where the combined total cost must be \leq \$100.

Linear inequalities allow us to build a mathematical model of this situation and understand the possible combinations of children and adults that fit within the budget constraints.

Graphing Inequalities

Graphing inequalities involves representing the possible solutions to inequalities on a coordinate plane. This helps visualize the relationships between variables and find the feasible solutions where the conditions of the inequalities are satisfied.

In this exercise, we graph two inequalities:

• For \( c + a \leq 20 \), the intercepts are (20,0) and (0,20). By plotting these points and shading below the line, we represent all the possible combinations of (c, a) that satisfy this condition.
• For \( 6c + 8a \leq 100 \), the intercepts are approximately (16.67,0) and (0,12.5). Again, we plot these points and shade below the line to show acceptable combinations of children and adults.

The area where the shaded regions of both inequalities overlap is known as the feasible region. This region represents all the (c, a) pairs that satisfy both inequality constraints simultaneously. By graphing the inequalities, we can easily determine which points are within the feasible region and thus meet the budget constraints for both matinee and evening tickets.

Feasible Region

The feasible region is a key concept when working with systems of inequalities. It represents the set of all possible solutions that satisfy all the inequalities in the system simultaneously.

• In our exercise, the feasible region is found by graphing both inequalities \( c + a \leq 20 \) and \( 6c + 8a \leq 100 \).
• The overlapping shaded area on the graph shows the combinations of children and adults that meet both budget constraints for matinee and evening tickets.
• Any point within this region represents a valid solution where the number of children and adults falls within budget limits.

For instance, both \( (c, a) = (9, 4) \) and \( (8, 5) \) are within the feasible region, meaning they are valid solutions to the problem. If a point lies outside this region, it means it does not satisfy one or both inequality constraints, and thus, it is not an acceptable solution for the given conditions.
The feasible region helps us quickly identify which combinations of variables work and which do not, making it a powerful tool in solving real-world problems involving multiple constraints.