Problem 35 - Algebra and Trigonometry with Analytic Geometry

Question

Inventory levels A store sells two brands of television sets. Customer demand indicates that it is necessary to stock at least twice as many sets of brand \(\mathrm{A}\) as of brand \(\mathrm{B}\). It is also necessary to have on hand at least 10 sets of brand \(B\). There is room for not more than 100 sets in the store. Find and graph a system of inequalities that describes all possibilities for stocking the two brands.

Step-by-Step Solution

Answer

1. \( x \geq 2y \) 2. \( y \geq 10 \) 3. \( x + y \leq 100 \). Graph the overlap.

Step 1: Define Variables

Let \( x \) represent the number of brand A televisions and \( y \) represent the number of brand B televisions. Our goal is to express the constraints in terms of these variables.

Step 2: Express Demand Constraint

According to the problem, the number of brand A television sets \( (x) \) must be at least twice the number of brand B sets \( (y) \). This means: \[ x \geq 2y \]

Step 3: Express Minimum Brand B Requirement

There must be at least 10 sets of brand B. Therefore: \[ y \geq 10 \]

Step 4: Express Space Constraint

The total number of television sets cannot exceed 100 units. Thus: \[ x + y \leq 100 \]

Step 5: Combine into a System of Inequalities

The constraints can be combined into the following system of inequalities: \[ x \geq 2y \] \[ y \geq 10 \] \[ x + y \leq 100 \]

Step 6: Graph the System of Inequalities

On a coordinate plane, plot each inequality:- For \( x \geq 2y \), shade the region to the right of the line \( x = 2y \).- For \( y \geq 10 \), shade above the line \( y = 10 \).- For \( x + y \leq 100 \), shade below the line \( x + y = 100 \).The solution is the intersection area of the shaded regions.

Key Concepts

Graphing Inequalities

When we talk about graphing inequalities, the goal is to represent inequality relationships visually on a coordinate plane. This visual representation makes it easier to understand which values satisfy the inequality.
To graph linear inequalities, you start by transforming the inequality into an equation. For example, if you have \( x \geq 2y \), you would first graph the line \( x = 2y \). This is your boundary line.

• Use a solid line if the inequality is \( \leq \) or \( \geq \) because the line itself is part of the solution.
• Use a dashed line for \( < \) or \( > \) since the line is not included in the solution.

After plotting the boundary line, shade the area representing the side where the inequality is true. For \( x \geq 2y \), you shade the region to the right of the line, as that's where \( x \) is greater than or equal to \( 2y \).
Consider each inequality separately first, then find the common shaded region after plotting all inequalities. This shared area is where all the conditions are satisfied simultaneously.

Systems of Inequalities

A system of inequalities consists of multiple inequalities considered together. Instead of finding individual solutions for separate inequalities, you're looking for a solution that satisfies all inequalities at once.
For instance, in the exercise, the system of inequalities is:

• \( x \geq 2y \)
• \( y \geq 10 \)
• \( x + y \leq 100 \)

To solve this system, follow these steps:- Graph each inequality on the same coordinate plane.- Identify and shade the region for each inequality.- The solution to the system is the overlapping, shaded region where all inequalities are true.
This overlapping region graphically represents the set of all possible solutions. In our example, this includes all combinations of television brands A and B that satisfy all the given constraints. Managing multiple constraints visually can be especially useful in making informed decisions when several factors are at play.

Linear Programming

Linear programming deals with optimizing a linear objective function, subject to a set of linear inequalities or constraints. In simpler terms, it's like finding the best possible outcome that meets all the given conditions.
Although the primary task in the given example involved graphing inequalities, it sets the stage for solving a linear programming problem. Here's a basic rundown on how linear programming would follow from such a setup:

• Define your objective function, like maximizing profit or minimizing cost.
• Use the system of inequalities as your constraints.
• Graphically, the feasible region (overlapping area from graphing the system) contains the optimal solution within its boundaries.

Linear programming identifies the point or points in this feasible region that best satisfy the objective function. Though more complex problems often require algebraic methods or specific software solutions, the foundation lies in understanding systems of linear inequalities and their graphical interpretations.