Problem 52 - Intermediate Algebra

Question

The set of integers between \(-7\) and 0

Step-by-Step Solution

Answer

The integers are -6, -5, -4, -3, -2, and -1.

Step 1: Understanding the Range

We need to find the integers that lie between -7 and 0. This means we should identify all whole numbers greater than -7 and less than 0.

Step 2: Identifying Integers within the Range

Integers are whole numbers without fractions. The integers that are greater than -7 and less than 0 are -6, -5, -4, -3, -2, and -1.

Step 3: Listing the Integers

Now, let's list all the integers from the previous step in order: -6, -5, -4, -3, -2, -1.

Key Concepts

Integer Range

When we talk about an "integer range," we refer to the sequence of integers that lie within a specific boundary, or interval, in this case being from \(-7\) to 0. To identify this range, we consider all integers that fall between these two points, but not including the boundaries themselves. Integers include both positive and negative whole numbers, essentially covering the entire spectrum with no fractions or decimal parts.
To determine the range between a couple of integers like \(-7\) and 0, consider all the step-by-step whole numbers that appear just after \(-7\) and just before 0 (in this case, less than 0). Hence, the integers formed in this range are sequential and uninterrupted, making them easy to list: \(-6\), \(-5\), \(-4\), \(-3\), \(-2\), and \(-1\).
Understanding integer ranges is essential when you are solving problems related to data sequencing or ordering specific values. It's a fundamental step in various mathematical processes, helping you find solutions or make decisions based on given parameters.

Whole Numbers

Whole numbers are a set of numbers that include all of the non-negative integers. These include the natural numbers like 0, 1, 2, 3, 4, and so on. It's important to note that whole numbers do not include fractions, decimals, or negative numbers. They start from zero (representing nothing) and increase by increments of one to infinity.
Whenever you encounter problems involving whole numbers, remember:

• They can be positive or zero.
• They are not subdivided or broken into fractional parts.
• They are adequate for counting objects or simple arithmetic operations like addition and subtraction.

Whole numbers are essential in representing counts that naturally occur in daily tasks and are fundamental to developing skills in basic arithmetic.

Negative Integers

Negative integers are the whole numbers less than zero. They are found on the left of zero on the horizontal number line, stretching infinitely in the negative direction. Understanding negative integers is a significant part of mastering the integer number line as they represent quantities less than zero, such as temperatures below freezing or owing money when your account balance dips below zero.
Key aspects of negative integers include:

• They are essential in solving problems involving debt or loss.
• Their magnitude increases as they move further from zero, but their value decreases (e.g., \(-1\) is greater than \(-5\)).
• They serve as a foundation for understanding more complex mathematical concepts such as absolute values and algebraic expressions.

Consequently, getting comfortable with negative integers provides a strong basis for advancing in mathematics and better interpreting real-world scenarios involving deficits or reductions.