Problem 316 - Elementary Algebra

Question

Jake does not want to spend more than \(\$ 50\) on bags of fertilizer and peat moss for his garden. Fertilizer costs \(\$ 2\) a bag and peat moss costs \(\$ 5\) a bag. Jake's van can hold at most 20 bags. (a) Write a system of inequalities to model this situation. (b) Graph the system. (c) Can he buy 15 bags of fertilizer and 4 bags of peat moss? (d) Can he buy 10 bags of fertilizer and 10 bags of peat moss?

Step-by-Step Solution

Answer

System of inequalities: \[\begin{cases} 2f + 5p \leq 50 \ f + p \leq 20 \ f \geq 0 \ p \geq 0 \end{cases}\]Yes, Jake can buy 15 bags of fertilizer and 4 bags of peat moss.No, Jake cannot buy 10 bags of fertilizer and 10 bags of peat moss.

Step 1 - Define Variables

Let’s define the variables: Let \( f \) be the number of bags of fertilizer and \( p \) be the number of bags of peat moss.

Step 2 - Set Up Inequalities for Cost

The total cost should not exceed \( \$50 \). Fertilizer costs \( \$2 \) per bag and peat moss costs \( \$5 \) per bag. Therefore, the inequality for cost is: \[ 2f + 5p \leq 50 \]

Step 3 - Set Up Inequality for Capacity

Jake’s van can hold at most 20 bags in total. Thus, the inequality for the van’s capacity is: \[ f + p \leq 20 \]

Step 4 - Non-negativity Constraint

Since Jake cannot purchase a negative number of bags, the third and fourth constraints are: \[ f \geq 0 \] \[ p \geq 0 \]

Step 5 - Write the System of Inequalities

The system of inequalities is: \[\begin{cases} 2f + 5p \leq 50 \ f + p \leq 20 \ f \geq 0 \ p \geq 0 \end{cases}\]

Step 6 - Graph the System

Graph the inequalities on a Cartesian plane. The feasible region will be the set of points that satisfy all inequalities.

Step 7 - Check Specific Points (Part (c) - 15 bags of fertilizer and 4 bags of peat moss)

Substitute \( f = 15 \) and \( p = 4 \) into the inequalities:i. \( 2(15) + 5(4) = 30 + 20 = 50 \leq 50 \)ii. \( 15 + 4 = 19 \leq 20 \)Both inequalities are satisfied, hence Jake can buy 15 bags of fertilizer and 4 bags of peat moss.

Step 8 - Check Specific Points (Part (d) - 10 bags of fertilizer and 10 bags of peat moss)

Substitute \( f = 10 \) and \( p = 10 \) into the inequalities:i. \( 2(10) + 5(10) = 20 + 50 = 70 \) (This does not satisfy \( 2f + 5p \leq 50 \))ii. \( 10 + 10 = 20 \leq 20 \)The first inequality is not satisfied, hence Jake cannot buy 10 bags of fertilizer and 10 bags of peat moss.

Key Concepts

Linear Inequalities

Linear inequalities are mathematical expressions that involve a linear function and inequality symbols such as <, >, ≤, or ≥. They are similar to linear equations but with inequalities instead of equal signs.

Here are some points to keep in mind about linear inequalities:

• They show a range of possible solutions, not just one specific value.
• They can be used to model various real-world scenarios like budget constraints or capacities.

In Jake's situation, we use linear inequalities to express the limits on his spending and the number of bags his van can carry. We have:
1. Cost Inequality: 2f + 5p ≤ 50
2. Capacity Inequality: f + p ≤ 20
3. Non-negativity Constraints: f ≥ 0 and p ≥ 0

Graphing Inequalities

Graphing inequalities helps us visualize the solutions that satisfy all the given conditions. To graph these inequalities, follow these steps:

• First, convert each inequality to an equation (replace ≤ or ≥ with =).
• Plot the line corresponding to each equation on a Cartesian plane.
• Determine which side of the line satisfies the inequality. You can do this by choosing a test point (usually (0,0) if it is not on the line).
• Shade the feasible side of the line; this represents all the points that satisfy the inequality.

When graphing Jake's system of inequalities, we get:
1. The line for 2f + 5p = 50, then shade below the line since 2f + 5p ≤ 50.
2. The line for f + p = 20, then shade below the line since f + p ≤ 20.

The feasible region is where the shaded areas overlap. This represents all the possible combinations of bags of fertilizer and peat moss Jake can buy within his constraints.

Feasibility Region

The feasibility region is a crucial concept in solving systems of inequalities. It represents all the possible solutions that satisfy all given inequalities.

The feasibility region is found by graphing all the inequalities and looking for the overlapping shaded regions.

• The corner points (vertices) of this region are where the boundary lines intersect.
• These points are often checked to find optimal solutions in problems involving optimization.

In Jake’s example, the feasible region is the area where the solutions for the budget and capacity constraints overlap. This is the set of all (f, p) pairs (number of fertilizer and peat moss bags) inside the intersecting shaded regions that satisfy:
1. 2f + 5p ≤ 50
2. f + p ≤ 20
3. f ≥ 0
4. p ≥ 0.

Any point within this region, including its boundaries, is a viable solution for Jake's purchase.

Algebraic Modeling

Algebraic modeling involves using algebraic expressions and equations to represent real-world problems mathematically. It helps in analyzing the relationships and constraints within the given problem.

Steps for algebraic modeling:

• Define the variables representing unknowns in the problem.
• Write down the inequalities based on the problem's constraints.
• Solve or graph the inequalities to find the feasible solutions.

In Jake's problem:

1. We defined variables: f (fertilizer bags) and p (peat moss bags).
2. Created inequalities based on budget (2f + 5p ≤ 50) and capacity (f + p ≤ 20).
3. Added non-negativity constraints (f ≥ 0, p ≥ 0).

Through algebraic modeling, we translated Jake's financial and space constraints into mathematical expressions. We then used these to determine feasible purchasing options.