Problem 317 - Elementary Algebra

Question

Reiko needs to mail her Christmas cards and packages and wants to keep her mailing costs to no more than \(\$ 500\). The number of cards is at least 4 more than twice the number of packages. The cost of mailing a card (with pictures enclosed) is \(\$ 3\) and for a package the cost is \(\$ 7\) (a) Write a system of inequalities to model this situation. (b) Graph the system. (c) Can she mail 60 cards and 26 packages? (a) Can she mail 90 cards and 40 packages?

Step-by-Step Solution

Answer

The system of inequalities is: \( 3x + 7y \leq 500 \) and \( x \geq 2y + 4 \). She can mail 60 cards and 26 packages but not 90 cards and 40 packages.

Step 1: Define the Variables

Let the number of cards be represented by the variable \( x \), and the number of packages be represented by \( y \).

Step 2: Set up the Cost Inequality

The total cost should not exceed \( \$ 500 \). The cost equation is \( 3x + 7y \leq 500 \).

Step 3: Set up the Number of Cards Inequality

The number of cards is at least 4 more than twice the number of packages: \( x \geq 2y + 4 \).

Step 4: Combine the Inequalities

The system of inequalities is: 1) \( 3x + 7y \leq 500 \) 2) \( x \geq 2y + 4 \).

Step 5: Graph the System

Plot the inequalities on a coordinate plane. The line \( 3x + 7y = 500 \) and the line \( x = 2y + 4 \) will be drawn, and the feasible region will be below them and to the right of \( x = 2y + 4 \).

Step 6: Check First Set of Values (60 cards and 26 packages)

Substitute \( x = 60 \) and \( y = 26 \) into both inequalities: 1) \( 3(60) + 7(26) = 180 + 182 = 362 \leq 500 \) 2) \( 60 \geq 2(26) + 4 = 52 + 4 = 56 \). Both conditions are true, so 60 cards and 26 packages are possible.

Step 7: Check Second Set of Values (90 cards and 40 packages)

Substitute \( x = 90 \) and \( y = 40 \) into both inequalities: 1) \( 3(90) + 7(40) = 270 + 280 = 550 ot\leq 500 \) 2) \( 90 \geq 2(40) + 4 = 80 + 4 = 84 \). The total cost exceeds \( \$ 500 \), so 90 cards and 40 packages are not possible.

Key Concepts

Understanding Linear Inequalities

Linear inequalities are mathematical expressions that show the relationship between two variables with inequality signs like \(\leq\), \(<\), \(\geq\), and \(>\). They look similar to linear equations, but their solutions cover a range of values that form a region on the graph rather than just a line.
For example, in the provided problem, one linear inequality is \(3x + 7y \leq 500\). This means that the total cost for mailing cards and packages cannot exceed \$500.

Another inequality is \(x \geq 2y + 4\), indicating that the number of cards must be at least 4 more than twice the number of packages. These inequalities together form a 'system of inequalities'.
Each inequality restricts the possible values for \(x\) and \(y\). Combining them helps determine which values meet both conditions. This is crucial for real-life problems like Reiko's mailing dilemma.

Graphing Inequalities

Graphing inequalities involves plotting the boundary lines derived from the inequalities on a coordinate plane. These lines split the plane into different regions. To graph the system of inequalities correctly, follow these steps:

First, graph each inequality line as if it were an equation. For instance, plot the line for \(3x + 7y = 500\).
• Choose points that satisfy the equality and draw the line.
• Do the same for the second inequality line \(x = 2y + 4\).

Next, determine which region satisfies the inequality. For the inequality \(3x + 7y \leq 500\), shade the region below and including the line, because \(\leq\) means 'less than or equal to'.
For the inequality \(x \geq 2y + 4\), shade the region to the right of the line, which represents 'greater than or equal to'.

The overlapping area of shading for both inequalities is called the feasible region. This region contains all possible solutions to the system. Reiko's solution lies within this feasible region.

Feasibility Region

The feasibility region is a crucial concept in solving systems of inequalities. It's the area on the graph where all the constraints (inequalities) overlap.
This region represents all potential solutions to the problem. For Reiko's mailing costs, the feasible region includes combinations of cards and packages that meet both her cost and quantity constraints.

To find the feasibility region, you:
• Graph each inequality on the same coordinate plane.
• Identify intersections and overlapping shaded areas.

In this example, the feasible region lies below the line \(3x + 7y = 500\) and to the right of the line \(x = 2y + 4\).
Checking specific points, like (60, 26), involves substituting these values back into the inequalities to see if they hold true. If both conditions are met, then the point is within the feasible region. This means Reiko can mail 60 cards and 26 packages without exceeding her budget and maintaining her quantity constraints.

Understanding the feasibility region helps solve many real-life problems by showing all possible solutions that meet given constraints efficiently.