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. Give two possible ways you could divide your hours between the two jobs.

Short answer

Two possible ways to divide your hours between the two jobs are: 15 hours as a cashier and 5 hours of babysitting OR 20 hours of babysitting and 0 hours as cashier.

Step-by-step solution

Set up the constraint for total hours

Let \( b \) represent the baby-sitting hours and \( c \) represent the cashier hours. Since the total work hours should not exceed 20 per week. So, we create the equation \( b+c \leq 20 \)

Set up the constraint for total earnings

Since the baby-sitting job pays 5 dollars per hour and the cashier job pays 6 dollars per hour, you earn 5b + 6c dollars per week. Therefore, the amount of money earned should be greater or equal to 90 dollars. Hence, \( 5b + 6c \geq 90 \)

Find two possible solutions

First we consider all cashier hours, c=15 and solve for b from the hour limit equation. \( b = 20 - c = 20 - 15 = 5 \). Checking this in the second inequality \( 5b + 6c = 5*5 + 6*15 = 25 + 90 = 115 \) which is greater than 90, thus it's a valid solution. For the next solution, we consider all babysitting hours, b=20 and solve for c from the hour limit equation. \( c = 20 - b = 20 - 20 = 0 \). These earnings are \( 5*20 + 6*0 = 100 \) which is also valid.

Key concepts

Algebraic Expressions

When it comes to algebra, one of the foundational elements is the understanding of algebraic expressions. These are combinations of letters (variables), numbers, and arithmetic operations like addition, subtraction, multiplication, and division. In the context of our example, the algebraic expressions are used to represent the number of hours you spend babysitting, denoted as \( b \), and cashiering, represented as \( c \).

The expression for the total earnings from both jobs is given by \( 5b + 6c \), which combines the hourly rates and the respective hours worked into a single equation. This equation makes it easier to manipulate and solve for unknown quantities based on given constraints. Understanding how to construct and work with algebraic expressions is imperative because it enables you to model and solve real-life scenarios, such as budgeting weekly expenses against your earnings from multiple jobs.

In practice, building an algebraic expression involves identifying the quantities involved, assigning variables to them, and then correctly applying arithmetic operations that reflect the relationships between those quantities. In our exercise, this is exemplified by translating hourly wages and total hours into a mathematical form that can be analyzed and solved.

Linear Equations

The linear equation is another critical concept within algebra, characterized by its first-degree variables (meaning the variables are not raised to any power higher than one) and its graph, which is a straight line. In the example used, the constraint for the total hours of work can be represented by the linear equation \( b + c \leq 20 \).

This particular kind of equation is called 'linear' because if we were to graph it, it would form a line. By graphing both of the given constraints, you can find their point of intersection, which represents a solution that satisfies both conditions. Linear equations are vital in solving problems related to budgeting, planning, and optimizing, as they provide a clear and straightforward method for determining possible solutions. Mastering linear equations aids students in understanding how to balance constraints and objectives in various situations, not only in mathematics but also in fields such as economics, physics, and engineering.

While working with linear equations, it is essential to remember that they often come with conditions, such as the ≤ or ≥ symbols in our problem, which indicate a range of possible solutions rather than a single solution point.

Inequalities

An inequality is like an equation, except instead of expressing equality, it shows a relationship where one side is greater or lesser in value than the other. In the context of our original exercise, two inequalities are used to represent the constraints on the number of work hours and the minimum earnings. The first inequality, \( b + c \leq 20 \), ensures that the combined hours for both jobs do not exceed 20 hours per week.

Inequalities are fundamental in modeling situations where there is a limit or minimum requirement - for instance, when there is a maximum number of hours you can work or a minimum amount of money you need to earn. Solving inequalities involves finding all possible values of variables that make the inequality true. This can be done through methods like substitution, as shown in the example, where the number of babysitting hours \( b \) is substituted into the total hours constraint to find the number of cashiering hours \( c \), ensuring the total earnings inequality \( 5b + 6c \geq 90 \) is satisfied.

Inequalities are not only crucial for algebraic problems but are also widely used in areas such as economics, business, and social sciences to make decisions based on limited resources, budgets, and other constraints. Understanding how to graph and solve them is an important skill, as they form the basis for understanding optimization problems and many other real-world applications.