Problem 33 - Elementary Algebra Within Reach

Question

A person plans to invest up to \(\$ 20,000\) in two different interest-bearing accounts, account \(X\) and account Y. Account \(\mathrm{X}\) is to contain at least \(\$ 5000\). Moreover, account \(\mathrm{Y}\) should have at least twice the amount in account \(X\). Write a system of linear inequalities that describes the various amounts that can be deposited in each account, and sketch the graph of the system.

Step-by-Step Solution

Answer

The final system of inequalities that describes the various amounts that can be deposited in each account is \(x + y \leq 20000\), \(x \geq 5000\), and \(y \geq 2x\). The graph of these system of inequalities will show the feasible region presenting the various amounts that can be held in the two accounts.

Step 1: Define Variables

Let's denote \( x \) as the amount of money (in dollars) to be deposited in account X and \( y \) as the amount of money (in dollars) to be deposited in account Y.

Step 2: Translate Words into Inequalities

Based on the problem, we have three conditions to consider: \n1. The total amount to invest is maximum $20,000, represented through the inequality \( x + y \leq 20000 \).\n2. There must be at least $5000 in account X, represented through the inequality \( x \geq 5000 \).\n3. Account Y should contain at least twice the amount in account X, represented through the inequality \( y \geq 2x \). This completes our system of linear inequalities.

Step 3: Arrange Inequalities

The final system of linear inequalities becomes: \n\n1. \( x + y \leq 20000 \) \n2. \( x \geq 5000 \) \n3. \( y \geq 2x \).

Step 4: Graph the Inequalities

The final step would be to sketch the inequalities on a graph. For this, x-axis can represent account X and y-axis can represent account Y. You start by plotting line for each inequality and then shading the area based on the inequality condition (less than or equal to, or greater than or equal to). The feasible region or solution will be where the shading overlaps for all conditions.

Key Concepts

Investment Constraints

Investment constraints refer to the limitations or conditions placed on how much money can be allocated into different accounts or investment options. In this exercise, you are dealing with two main constraints.
You need to understand:

• There is a maximum total investment limit, which is set at $20,000.
• The minimum investment required in account X is $5,000.
• Account Y must contain at least twice the amount of account X.

These constraints help investors allocate funds optimally while adhering to required conditions both for risk management and return optimization purposes.
By defining constraints clearly, you ensure no more than the available amount is invested and that each account achieves its goals, whether growth or liquidity.

Graphing Inequalities

Graphing inequalities is about visually representing the conditions of a system on a coordinate plane. This can help you understand the feasible region, which is the set of all possible solutions that satisfy all inequalities simultaneously.
In this exercise, you graph inequalities for three conditions:

• The total investment: the line from the inequality \(x + y \leq 20000\) is plotted and represents how much can be invested collectively.


• Minimum in account X: the vertical line from the inequality \(x \geq 5000\) indicates that investment in account X should not fall below $5,000.


• Minimum in account Y: plotted from \(y \geq 2x\) showing that money in account Y must be at least twice that in account X.

As each line is drawn, the area satisfying each condition is shaded. The resulting overlap, known as the feasible region, shows all possible combinations of investments that meet all constraints.

Defining Variables

Defining variables is the first and crucial step in converting a word problem into a mathematical framework. Variables act as placeholders that can represent unknown quantities.
In this particular exercise, you encounter two variables:

• Let \(x\) be the amount of money in account X.
• Let \(y\) be the amount of money in account Y.

Choosing appropriate symbols or letters keeps the problem simple and comprehensible.
By defining your variables clearly at the start, calculations become straightforward, avoiding confusion as you translate the verbal conditions of the problem into mathematical inequalities.
It ensures that each part of the problem aligns with the corresponding variable, aiding in the clarity and streamlined problem-solving.

Inequality Translation

Inequality translation involves converting verbal statements or conditions into mathematical expressions. It's essential for setting up equations that describe the problem effectively.
From the problem context, you have:

• The maximum total investment condition: expressed as \(x + y \leq 20000\).


• Minimum in account X: represented by \(x \geq 5000\).


• Account Y relationship to X: conveyed through \(y \geq 2x\).

Learning to translate verbal statements into inequalities is crucial as it allows us to precisely define mathematical relationships and constraints. This process sets the stage for further analysis, such as solving for feasible regions or optimal solutions in various mathematical scenarios. It's an invaluable skill in understanding and formulating problems from real-world contexts into solvable mathematical equations.