Problem 57 - College Algebra

Question

A person with no more than 15,000 dollars to invest plans to place the money in two investments. One investment is high risk, high yield; the other is low risk, low yield. At least 2000 dollars is to be placed in the high-risk investment. Furthermore, the amount invested at low risk should be at least three times the amount invested at high risk. Find and graph a system of inequalities that describes all possibilities for placing the money in the high- and low-risk investments.

Step-by-Step Solution

Answer

The system of inequalities for all possible investments is H + L ≤ 15000, H ≥ 2000 and L ≥ 3H. Graphing these inequalities will give a feasible region which shows all the valid allocations of the money between the high-risk and low-risk investments.

Step 1: Understanding the Problem and Formulating the Inequalities

From the problem, a person has at most $15,000 to invest. Therefore, the sum of investments in both high-risk (H) and low-risk (L) should not exceed $15,000: H + L ≤ 15000. The problem also states that at least $2000 should be placed in high-risk investment: H ≥ 2000. Lastly, the low-risk investment should be at least three times the amount in high-risk: L ≥ 3H.

Step 2: Graphing the Inequalities

Representing the high-risk investment on the x-axis and the low-risk on the y-axis, plot the lines H + L = 15000, H = 2000, and L = 3H on a graph paper. Remember that areas below the lines represent 'less than or equal to' inequalities while areas above represent 'more than or equal to' inequalities. Therefore, shade the appropriate regions to indicate the inequalities: below for H + L ≤ 15000, above for H ≥ 2000, and above for L ≥ 3H. The feasible region will be the common region that satisfies all these conditions.

Key Concepts

Inequality Graphing

Inequality graphing is a foundational technique in algebra that helps visualize and solve problems involving inequalities. Imagine drawing a boundary line for each inequality, which represents all the points that turn the inequality from '<' to '='. When you're dealing with a system of inequalities, like in our investment example, the solution is not just a single point or line but an entire region that satisfies all the inequalities at once.

Start by sketching the x and y axes, where each axis corresponds to one investment. For our example, let's say the x-axis is the high-risk investment (H) and the y-axis is the low-risk investment (L). Then, plot the boundary lines based on the inequalities identified:

• The line representing the total investment limit, which in our case is the line defined by the equation H + L = 15,000.
• The line representing the minimum high-risk investment, or H = 2,000.
• The line for the constraint that the low-risk investment should be at least three times the high-risk investment, so L = 3H.

Here's the key: shading. Below the line H + L = 15,000, because you can't exceed the total amount. Above the line H = 2,000 because you need to invest at least that much in high-risk. Finally, above the line L = 3H because the low-risk investment needs to be three times the high-risk amount. The feasible region, where all shaded areas overlap, shows all the valid investment options.

Investment Strategies Algebra

Investment strategies can often be analyzed using algebraic models, which include a system of linear inequalities. Each inequality represents a constraint or a rule guiding where money could be invested. In the context of our example with high-risk and low-risk investments, algebra helps outline a plan that meets several criteria simultaneously, ensuring that the investor's decisions adhere to their risk tolerance and amount available for investment.

Investment strategy formulation can be summarized in a few steps:

1. Establish the maximum and minimum amounts to be allocated in each investment type based on personal preferences, risk assessment, and financial goals.
2. Translate these into linear inequalities, where the constraints are represented by algebraic expressions.
3. Solve these inequalities individually, and then look at the system as a whole to find a common solution that satisfies all conditions.

The key benefit of translating investment strategies into algebraic terms is the clear visual representation it provides. It allows an investor to quickly assess viable options and make informed decisions. In this exercise, it's clear that any strategy must include a minimum of \(2,000 in high-risk and at least triple that amount in low-risk, without exceeding an overall investment of \)15,000.

Linear Inequalities

Linear inequalities are similar to linear equations but express a range of solutions instead of one exact solution. The inequality symbols (<, >, ≤, ≥) indicate the relationship between two algebraic expressions. In real-world scenarios like our investment example, linear inequalities are used to model constraints and limitations.

Understanding and solving linear inequalities involve a few essential steps:

• Identify the inequality symbols and what they represent in the context of the problem.
• Rearrange the inequality into a standard form, when necessary, to clearly identify the slope and y-intercept, which are key to graphing.
• Graph each inequality by first sketching the corresponding equality as a solid or dashed line – dashed for '<' or '>', solid for '≤' or '≥' – and then shading the appropriate side of the graph.
• Determine the solution by finding where the shaded regions of all inequalities overlap, known as the feasible region.

In our exercise, we've been introduced to three main linear inequalities which define the boundaries of the investment funds. They lead to creating a graph where all acceptable investment options lie within a certain area, providing a powerful visual aid for decision-making.