Problem 38 - Algebra 1

Question

Use the following information. Suppose you can work a total of no more than 20 hours per week at your two jobs. Baby-sitting pays 5 dollar per hour, and your cashier job pays 6 dollar per hour. You need to earn at least 90 dollar per week to cover your expenses. Write a system of inequalities that shows the various numbers of hours you can work at each job. Graph the result.

Step-by-Step Solution

Answer

The system of inequalities is \( x+y \leq 20 \) and \( 5x + 6y \geq 90 \). These inequalities represent the constraints on your time and earnings from the two jobs, respectively.

Step 1: Determine the Variables

Let \( x \) represent the number of hours spent baby-sitting, and \( y \) represent the number of hours spent as a cashier.

Step 2: Write the Constraints as Inequalities

You can work a total of no more than 20 hours per week. Therefore the first inequality is \( x+y \leq 20 \). The second inequality is based on your earnings. Baby-sitting pays 5 dollar per hour and your cashier job pays 6 dollar per hour, and you need to earn at least 90 dollar per week. This gives us the inequality: \( 5x + 6y \geq 90 \).

Step 3: Graph the Inequalities

Graph both inequalities \( x+y \leq 20 \) and \( 5x + 6y \geq 90 \) on the same system of coordinates. The feasible region represents the different combinations of hours you can work at each job that will meet your needs.

Key Concepts

Graphing Inequalities

When it comes to understanding system of inequalities, graphing them is a powerful visual tool. Imagine a coordinate plane, a grid with an x-axis and a y-axis. To graph an inequality, such as the example given where we say the number of hours baby-sitting plus the number of hours working as a cashier must be less than or equal to 20 (\(x + y \textless = 20\)), we start by treating the inequality as if it were an equation (\(x + y = 20\)) and draw the line that represents it.

Once this line is drawn, we then have to consider the inequality part. Since we want the sum of hours to be less than or equal to 20, we shade the area below the line — because any point in this area represents a possible combination of hours that satisfies the inequality. Similarly, for the inequality representing earnings (\( 5x + 6y \textgreater = 90\text{ dollars}\)), the region where you make at least $90 is shaded. The intersection of the two shaded regions shows the range of possible hours you can work at both jobs while meeting your constraints. These regions are crucial as they represent all the solutions to the system of inequalities. To enhance understanding, it's helpful to use different colors or patterns when graphing multiple inequalities.

Linear Programming

Linear programming involves finding the optimal solution from a set of possible solutions that satisfies a given set of restrictions or constraints. In the example problem, we essentially engaged in a basic form of linear programming. Our objective is to maximize or minimize some quantity - in this case, it's the number of hours we can work within the given constraints. With the inequalities \(x+y \textless = 20\text{ hours}\) and \(5x + 6y \textgreater = 90\text{ dollars}\), we are constrained by both time and earnings.

This situation is perfectly suited for graphing, as the feasible region visualized on the graph encompasses all the solutions that satisfy the system of inequalities. In more complex problems, linear programming will make use of techniques, such as the Simplex Method, to find the optimal solution at the vertices of the feasible region. However, for this type of problem, simply identifying the feasible region on a graph provides valuable insights into possible solutions.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables (like \(x\) and \(y\)), and operation symbols. They represent real-world quantities without an equals sign, unlike equations. In the context of the example problem, \(x\) and \(y\) are algebraic variables that represent hours you might work at either job. The expressions \(5x\) and \(6y\) convert these hours into dollars, showing the total earnings from each job.

It's essential to interpret these algebraic expressions properly. The expression \(x + y\) is simple; it combines the hours from both jobs without converting into dollars. However, the expressions for earnings, \(5x\) and \(6y\), reflect different hourly rates and thus shouldn't be combined directly — they contribute separately to the inequality that captures the total earnings required each week. Understanding each term in your expression helps with setting up and solving the problems effectively.