Problem 70 - Fundamentals of Mathematics

Question

Perform each operation. $$ \frac{-120}{-|2|} $$

Step-by-Step Solution

Answer

The result of the operation is 60.

Step 1: Evaluate the Absolute Value

The expression involves an absolute value within a denominator. First, resolve the absolute value of 2. Since the absolute value of any number is its distance from zero, ignoring the sign, we have \[ |2| = 2 \].

Step 2: Insert the Absolute Value into the Denominator

Now substitute the evaluated absolute value back into the expression, replacing \(-|2|\) with \(-2\). Thus, the expression becomes \[ \frac{-120}{-2} \].

Step 3: Perform the Division

Now perform the division of the numerator by the denominator. Dividing \(-120\) by \(-2\) results in a positive quotient because the division of two negative numbers is positive. Calculate \[ \frac{-120}{-2} = 60 \].

Key Concepts

Understanding Rational Expressions

Rational expressions are a fundamental part of algebra. They are fractions where both the numerator and the denominator are polynomials. In simpler terms, a rational expression can look like a simple fraction that you've encountered before, but it often involves variables in addition to numbers. In our given exercise, the rational expression is noteworthy because it involves divisions using integers. It's important to remember that the division of integers follows the same rules as any other division.

• For example, just like numbers, a singular variable can be treated similarly.
• These expressions can be simplified or manipulated through standard mathematical operations like addition, subtraction, multiplication, and division of polynomials.

By understanding the basic structure of rational expressions, you foster a better comprehension of more complex algebraic expressions and the mathematical relationships within them.

Performing Numerical Operations

Numerical operations refer to the basic arithmetic functions we use daily: addition, subtraction, multiplication, and division. In the context of the exercise, we are primarily interested in subtraction and division. Let's break it down:

• Subtraction takes place when converting the absolute value term in the denominator.
• Division happens when the simplified rational expression is evaluated, as shown in the solution.

Understanding and mastering these operations is crucial as they form the basis for more complex calculations. Numerical operations with absolute values can sometimes seem tricky because absolute value affects only the sign, not the number's magnitude. So, even when the numbers are negative, these operations follow the same basic principles, ensuring consistent results.

The Process of Integer Division

Integer division is specifically the division of whole numbers. A key point to remember is that when dividing two negative numbers, the result is a positive integer. In our problem, both (-120) and (-2) are integers.

• The division of (-120) by (-2) is a straightforward process as negative divided by negative yields positive.
• This result can be thought about like reversing a number line when twice flipped, leading you to a positive result.
• Integer division is distinct in how it handles remainders, but in this problem, there’s none, simplifying the calculation.

Keep these steps in mind, as integer division forms a vital part of numerical problem-solving. It becomes second nature once you grasp the concept of positive outcomes arising from negative divisions.