Problem 61 - Elementary Algebra

Question

Due to a slowdown in the industry, a Silicon Valley computer company finds itself in debt $$\$ 2,400,000$$. Use a bsolute value notation to describe this company's debt.

Step-by-Step Solution

Answer

Answer: The absolute value notation for the company's debt is $$|-2,400,000|= 2,400,000$$.

Step 1: Understand the Concept of Absolute Value

Absolute value is the distance of a number from zero on the number line. It is denoted by two vertical bars surrounding the number (Example: $$|x|$$). The absolute value of a positive number or zero is the number itself, and the absolute value of a negative number is the opposite of the number.

Step 2: Represent the Company's Debt as a Negative Number

As the company is in debt, we need to represent the debt amount as a negative number. In this case, the company is in debt by $2,400,000, so the negative representation of this debt is $$-2,400,000$$.

Step 3: Write the Absolute Value Notation for the Debt

Now that the company's debt is represented as a negative number, we can use absolute value notation to describe the magnitude of the debt. In this case, the absolute value of the company's debt is: $$|-2,400,000|$$.

Step 4: Simplify the Absolute Value Notation

To find the absolute value of a negative number, we simply take the opposite of the number as mentioned earlier. So the absolute value of the company's debt is: $$|-2,400,000| = 2,400,000$$.

Key Concepts

Negative Numbers

A negative number is simply a number less than zero. On the number line, these numbers appear to the left of zero. Negative numbers represent any value that is less than nothing, such as

• temperature readings below zero
• or a debt that needs to be paid off.

They are crucial in mathematics as they help describe real-world situations where losses, reductions, or decreases occur. The mathematical symbol for negative numbers is a minus sign (−). For example, if you have a negative balance or a debt of \(2,400,000, it could be represented as \(-2,400,000\). This means you owe \)2,400,000. Understanding that a negative number implies owing or lacking something is key.

Representing Debt

Debt often means owing money to someone—like a loan or a credit card bill. In arithmetic and algebra, debt is typically represented by negative numbers. When accountants and financial experts speak of debt, they frequently describe it using negatives because it clearly indicates that an amount needs to be paid.
For instance, if a company is in debt by $2,400,000, this amount is shown as \(-2,400,000\). The negative sign effectively communicates that the company needs to pay back this amount.

• Positive numbers equate to having money or assets.
• Negative numbers indicate debts or liabilities.

So, when debt is represented by negative numbers, it simplifies the understanding of financial health.

Mathematical Notation

Mathematical notation is a system of symbols and signs that convey mathematical ideas clearly and efficiently. One important notation is the absolute value, which shows how far a number is from zero on the number line, regardless of direction. It's like asking, "How big is the number, ignoring if it is negative or positive?"
The absolute value is notated with two vertical bars, e.g., \(|x|\). For a debt of \(-2,400,000\), the absolute value notation would be \(|-2,400,000|\), meaning the magnitude of the debt is actually $2,400,000 without considering it is owed.
Some other common mathematical notations include

• \( + \) for addition
• \( - \) for subtraction
• \( \times \) or \( \cdot \) for multiplication
• \( \div \) for division.

Using these tools, mathematicians and students can express complex ideas concisely and accurately.