Problem 84 - College Algebra Essentials

Question

A person with no more than \(\$ 15,000\) 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 \(\$ 2000\) 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 inequalities representing the investment problem are: \(x \geq 2000\), \(y \geq 3x\), and \(x + y \leq 15000\). The feasible region is the shaded area in the graph that satisfies all three inequalities.

Step 1: Identify and define variables

Let's define \(x\) to be the amount of money invested in the high-risk investment and \(y\) to be the amount of money invested in the low-risk investment.

Step 2: Formulate the inequalities

From the problem we know that at least \(\$2000\) must be invested in the high-risk investment, which translates to the inequality \(x \geq 2000\). We also know that the amount invested at low risk should be at least three times the amount invested at high risk, resulting in the inequality \(y \geq 3x\). Finally, the total investment shouldn't exceed \(\$15000\), which gives us the inequality \(x + y \leq 15000\).

Step 3: Graph the inequalities

These inequalities can be graphed on a coordinate plane, with \(x\) representing the high-risk investments on the x-axis, and \(y\) representing the low-risk investments on the y-axis. Draw a vertical line at \(x = 2000\), a line with slope of 3 through the origin and a line with negative slope through the point where \(x = 15000\) and \(y = 0\). Shade the area of the graph that fulfills all three inequalities.