Problem 50 - Intermediate Algebra

Question

Boat Sales. Dry Boat Works wholesales aluminum boats for $$ 800\( and fiberglass boats for \) 600 .\( Northland Marina wants to order at least \)2,400\( worth but no more than \)4,800\( worth of boats from them. Use the illustration to graph a system of inequalities that will show the possible combinations of the number of aluminum boats \)(x)\( and the number of fiberglass boats ( \)y$ ) that can be ordered. Give two possible solutions. (GRAPH CANNOT COPY)

Step-by-Step Solution

Answer

Possible solutions are (3,2) and (6,0).

Step 1: Define the Variables

Let \( x \) represent the number of aluminum boats, and \( y \) represent the number of fiberglass boats. The prices are \(800 for aluminum boats and \)600 for fiberglass boats.

Step 2: Establish the Inequalities

You need to write inequalities for the total cost of the boats. The total wholesale cost can be expressed by the equation: \( 800x + 600y \). Northland Marina wants to order at least \(2,400 worth, so the inequality will be \( 800x + 600y \geq 2400 \). They also do not want to exceed \)4,800, so the second inequality is \( 800x + 600y \leq 4800 \).

Step 3: Simplify the Inequalities

Divide each term in the inequalities by 200 to simplify them. For the first inequality: \( 800x + 600y \geq 2400 \) becomes \( 4x + 3y \geq 12 \). For the second inequality: \( 800x + 600y \leq 4800 \) becomes \( 4x + 3y \leq 24 \).

Step 4: Solve for y in Each Inequality

Rearrange each simplified inequality to solve for \( y \) in terms of \( x \). From \( 4x + 3y \geq 12 \), we get \( 3y \geq 12 - 4x \) and then \( y \geq \frac{12 - 4x}{3} \). From \( 4x + 3y \leq 24 \), we get \( 3y \leq 24 - 4x \) and then \( y \leq \frac{24 - 4x}{3} \).

Step 5: Graph the Inequalities

Plot the equations \( y = \frac{12 - 4x}{3} \) and \( y = \frac{24 - 4x}{3} \) on a coordinate graph. The area between these lines, including the lines themselves, represents the solution set for \( x \) and \( y \). Be sure to shade the region between the lines since the inequalities are inclusive.

Step 6: Identify Two Possible Solutions

Find integer solutions \( (x, y) \) that satisfy both inequalities. Possible solutions within the shaded area could be \( (3,2) \) and \( (6,0) \). Check: For \( (3,2) \), \( 800(3) + 600(2) = 3600 \); for \( (6,0) \), \( 800(6) + 600(0) = 4800 \), both are within the specified range.

Key Concepts

Understanding Algebra in This Context

Algebra is a branch of mathematics where you use symbols and letters to represent numbers in equations and expressions. In this exercise, algebra comes into play as you define variables and set up equations to model a real-world problem. Here, the variables are used to represent the number of aluminum and fiberglass boats - let \( x \) stand for aluminum boats and \( y \) for fiberglass boats.

• Price Multiplication: Multiply these variables by their respective prices (\( 800 \) for aluminum and \( 600 \) for fiberglass) to calculate the total cost expression: \( 800x + 600y \).
• Creating Inequalities: Then, you form inequalities to model the sales constraints.
• Simplification: Simplifying these inequalities aids in solving them more effectively.

By engaging with these algebraic processes, you're able to transform the scenario into a set of mathematical expressions that can be analyzed and solved.

Linear Inequalities Explained

Linear inequalities are similar to linear equations but instead of expressing equality, they show a range of possible solutions. These are equations that use inequalities like \( \leq \) or \( \geq \) instead of \( = \).

• Formulating Inequalities: In the exercise, the inequality \( 800x + 600y \geq 2400 \) represents the company's wish to spend at least \( 2,400 \).
• Upper limits are shown by \( 800x + 600y \leq 4800 \), meaning they don't want to exceed \( 4,800 \).
• Solving for \( y \): You then rearrange these to express \( y \) in terms of \( x \), simplifying the inequalities to \( y \geq \frac{12 - 4x}{3} \) and \( y \leq \frac{24 - 4x}{3} \).

These inequalities maintain the relationship between the variables within specific boundaries, setting the stage for graphing the solution.

Graphing Inequalities for Solutions

Graphing inequalities is a practical way to visualize the solution set to a system of inequalities. In this situation, it represents the valid combinations of aluminum \( (x) \) and fiberglass boats \( (y) \) orders.

• Drawing Lines: Graph the lines \( y = \frac{12 - 4x}{3} \) and \( y = \frac{24 - 4x}{3} \) on the coordinate plane to mark the boundary of solutions.
• Shading the Solution Area: The region between these two lines (including the lines), where both inequalities are satisfied, is shaded.
• Finding Solutions: Look for integer coordinates in this shaded area that represent feasible solutions. For example, \((3, 2)\) and \((6, 0)\) are valid points.

This graphical method not only provides a clear visual representation allowing you to dynamically find solutions but also helps you understand relationships between different variables in algebra.