Problem 47 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(47-76,\) perform the indicated division or state that the expression is undefined. $$\frac{12}{-4}$$

Step-by-Step Solution

Answer

-3

Step 1: Understand the Question

We are asked:
In Exercises \(47-76,\) perform the indicated division or state that the
expression is undefined.
$$\frac{12}{-4}$$

Step 2: Recall the Definition

We recall the relevant mathematical definition or concept.

Step 3: State the Answer

-3

Key Concepts

Numerator

In a division problem, the numerator is the top part of the fraction. It represents the quantity that is being divided by another number. In the expression \(\frac{12}{-4}\), the number 12 is the numerator.
Think of the numerator as the total amount you have before it is shared or divided. The goal is to distribute this number across the parts indicated by the denominator.
When performing the division, the numerator is where you start – it's what will eventually be evenly divided.

Denominator

The denominator in a division problem is the bottom part of the fraction. It shows how many parts the numerator is divided into. In \(\frac{12}{-4}\), the denominator is \(-4\).
Handling the denominator is crucial as it dictates the division's outcome. If the denominator is zero, the expression becomes undefined, as dividing any number by zero is not possible.
In our example, because the denominator is \(-4\), it tells us to separate the quantity 12 into 4 negative parts.

Negative Numbers

Negative numbers can seem tricky at first, but they're simply numbers less than zero. They are used to represent values below zero like debts or temperatures. In division, they follow specific rules.
When dividing a positive number by a negative number, the result is negative. Similarly, dividing two negative numbers gives a positive result. So, in the expression \(\frac{12}{-4}\), we divide a positive by a negative, leading to \(-3\).
These rules help maintain consistency within arithmetic operations and are crucial for solving problems correctly.

Arithmetic Operations

Arithmetic operations include basic mathematical operations such as addition, subtraction, multiplication, and division. These are fundamental for solving equations and handling numbers in different ways.
Division is one of these operations, represented by fractions, like the one in our expression \(\frac{12}{-4}\).
Understanding arithmetic allows us to manipulate numbers efficiently and figure out values for more complicated mathematical problems. Mastering these operations helps in everyday calculations and in developing a strong mathematical foundation.