Problem 36 - Algebra A Combined Function

Question

Tony Noellert budgets his time at work today. Part of the day he can write bills; the rest of the day he can use to write purchase orders. The total time available is at most 8 hours. Less than 3 hours is to be spent writing bills. a. Write a system of inequalities to describe the situation. (Let \(x=\) hours available for writing bills and \(y=\) hours available for writing purchase orders.) b. Graph the solutions of the system.

Step-by-Step Solution

Answer

The system of inequalities is: \( x+y \leq 8 \), \( x < 3 \), \( x \geq 0 \), and \( y \geq 0 \).

Step 1: Defining the Variables

We define the variables: Let \( x \) represent the number of hours Tony spends writing bills, and \( y \) represent the number of hours he spends writing purchase orders.

Step 2: Creating the Inequalities for Time Constraints

Since the total available time is at most 8 hours, the inequality for this constraint is: \( x + y \leq 8 \). Less than 3 hours is to be spent on writing bills, which can be represented as \( x < 3 \). Since time cannot be negative, we also have the non-negative constraints: \( x \geq 0 \) and \( y \geq 0 \).

Step 3: Writing the System of Inequalities

Based on the constraints mentioned, the system of inequalities can be written as: \[ \begin{align*} x + y & \leq 8, \ x & < 3, \ x & \geq 0, \ y & \geq 0. \end{align*} \]

Step 4: Graphing the Inequalities

To graph the system, first plot the line \( x + y = 8 \). The solution region will be the area below this line. Secondly, plot the vertical line at \( x = 3 \). Since \( x < 3 \), the solution is to the left of this line. Include the axes with \( x \geq 0 \) and \( y \geq 0 \). The feasible region is the intersection of these areas.

Key Concepts

Graphing Inequalities

Graphing inequalities is similar to graphing equations, but instead of a line representing the solution, you have a region. In Tony's example, we graph two major inequalities: \(x + y \leq 8\) and \(x < 3\). The line \(x + y = 8\) is a boundary for the total time spent, and the solution region is below this line.
For \(x < 3\), you'll draw a vertical line at \(x = 3\) and shade to the left to indicate that time spent on bills should be less than 3 hours. The axes represent the non-negativity conditions \(x \geq 0\) and \(y \geq 0\).
When plotting these on a graph, look for where the shaded regions overlap. This intersection zone is the solution to the system, showing the feasible times Tony can divide between tasks.

Constraints in Algebra

Constraints in algebra help us define what is possible within a mathematical model. They are rules or limits to what a solution can be. In the exercise with Tony, constraints specify how much time he can dedicate to certain tasks.
Algebraic constraints are expressed as inequalities which tell us about maximum, minimum, or both limits. For example, \(x + y \leq 8\) sets a maximum combined working time of 8 hours.

• \(x < 3\) ensures a strict upper bound on time writing bills.
• \(x \geq 0\) and \(y \geq 0\) confirm no negative working hours.

These constraints determine Tony's feasible schedule by outlining boundaries and allowable combinations of time spent on each task.

Linear Inequalities

Linear inequalities involve expressions where two variables have a linear relationship, represented by an inequality sign such as \(<\), \(>\), \(\leq\), or \(\geq\). Tony's case includes several linear inequalities that represent his time management limits.
The inequality \(x + y \leq 8\) shows the constraint of total working hours, while \(x < 3\) limits the time on a specific task. Linear inequalities can represent conditions like maximum capacity or available resources.

• When graphing, each inequality splits the plane into two regions.
• The boundary line marks where the inequality changes from true to false.

Identifying the true region gives insight into the various combinations of values that satisfy all conditions.

Real-World Application of Algebra

Algebra is not just theoretical; it has practical uses, especially in planning and resource management. Tony's exercise is a typical real-world example of how algebra helps allocate time effectively.
Using systems of inequalities, Tony can find the optimal way to split his 8-hour work day. This technique isn't limited to individual tasks but extends to business logistics, like resource allocation or scheduling.

• Inequalities model constraints on resources available.
• They aid in optimizing operations without exceeding limits.

By understanding algebra and inequalities, you learn to balance different requirements efficiently, making informed decisions in various life scenarios.