Problem 337 - Elementary Algebra

Question

Find your last month's phone bill and the hourly salary you are paid at your job. (If you do not have a job, use the hourly salary you would realistically be paid if you had a job.) Calculate the number of hours of work it would take you to earn at least enough money to pay your phone bill by writing an appropriate inequality and then solving it.

Step-by-Step Solution

Answer

Work at least \frac{B}{S}\ hours to pay your phone bill.

Step 1 - Determine Phone Bill Amount

Find the total amount of last month's phone bill. Let's call this amount \(B\).

Step 2 - Determine Hourly Salary

Determine the hourly salary you are paid at your job. Let's call this amount \(S\).

Step 3 - Write the Inequality

To find out how many hours you need to work, write the inequality \(S \cdot H \geq B\), where \(H\) represents the hours worked.

Step 4 - Solve for Hours Worked

Isolate \(H\) in the inequality \(S \cdot H \geq B\) by dividing both sides by \(S\). Hence, you get \(H \geq \frac{B}{S}\).

Step 5 - Round Up the Hours (if necessary)

If \(H\) is not a whole number, round up to the nearest whole hour since you can't work a fraction of an hour in most jobs.

Key Concepts

Linear Equations

Linear equations are among the simplest types of equations in algebra. They are called 'linear' because they graph as straight lines. A linear equation is one in which the highest power of the variable is 1. For example, in the inequality you wrote to solve your phone bill problem, such as: \[ S \times H \geq \ B \], The equation is linear because the variable \( H \) (hours worked) has a power of 1. When we solve this inequality, we are looking for the range of values that make the inequality true. In this context, it means finding how many hours you need to work to cover your expenses.

Variables and Constants

In algebra, it's essential to understand the difference between variables and constants. This distinction helps in forming and solving equations effectively. A variable represents an unknown value that can change or vary. In our example: \( S \) and \( H \) are variables. \( S \) is the hourly salary and \( H \) is the number of hours worked. A constant is a fixed value that does not change. For this problem, the phone bill amount \( B \) is a constant because it's a set amount you need to pay. Identifying what represents a variable and what represents a constant is crucial when forming and manipulating equations and inequalities. In real-life applications, determining these can make solving problems much more straightforward.

Real-World Applications

Solving inequalities and linear equations is not just a classroom activity; it has practical real-world applications. Let's consider the phone bill example. By translating this situation into a mathematical inequality, you can determine how to budget your time and money more effectively. Here's how the steps apply in the real world:

• Determine Phone Bill Amount: This represents identifying the exact cost of your obligations or needs.
• Determine Hourly Salary: Understand how much you are earning per hour.
• Write the Inequality: Formulating the inequality \( S \cdot H \geq \ B \) models your financial situation mathematically.
• Solve for Hours Worked: Isolating \( H \) tells you the minimum hours you need to work to meet your expenses.
• Rounding Up: Often, real-life constraints require working full hours, making rounding up essential.

These steps replicate many budgeting and financial planning methods you'll encounter beyond school. This practical approach helps in decision-making and efficient time management.