Problem 15 - Fundamentals of Mathematics

Question

What numbers can replace \(x\) so that each statement is true? \(-8

Step-by-Step Solution

Answer

The numbers are -7, -6, -5, -4, -3, and -2.

Step 1: Understanding the Inequality

The problem states that \(-8 < x \leq -2\). This inequality means that \(x\) should be greater than \(-8\) but less than or equal to \(-2\).

Step 2: Identifying Whole Numbers

A whole number is a non-negative integer. However, in mathematics, the term 'whole number' can sometimes be used for non-negative integers whereas the criteria given here imply negative values. We'll assume the problem means integers, which include negative and positive whole numbers. Hence, \(x\) must be a whole number within the specified inequality.

Step 3: Finding Valid Values for \(x\)

The values for \(x\) need to satisfy both conditions: they must be whole numbers (integers) and must fit within the inequality \(-8 < x \leq -2\). Checking integers within the range, the valid whole numbers are: \(-7, -6, -5, -4, -3,\) and \(-2\).

Step 4: Listing the Possible Values

After identifying the integers within the specified range, we finalize that the numbers which can replace \(x\) to satisfy both statements of the problem are \(-7, -6, -5, -4, -3,\) and \(-2\).

Key Concepts

Integers

Integers are a fundamental concept in mathematics. They include all whole numbers, both positive and negative, and zero. Understanding integers is crucial for grasping how inequalities work with various kinds of numbers. Let's break down some characteristics of integers:

• Integers are the set of numbers that include positive numbers (e.g., 1, 2, 3), negative numbers (e.g., -1, -2, -3), and zero (0).
• Unlike fractions or decimals, integers are whole units without any fractional part.
• They play a crucial role in many areas of mathematics, including algebra and number theory.

When dealing with inequalities involving integers, such as \(-8 < x \leq -2\), the integers that satisfy the inequality are the complete whole numbers within the specified range. They do not include numbers like \(-8\) because it's not greater than \(-8\), but they do include \(-2\) because \(-2\) matches the condition of being less than or equal to \(-2\). Understanding integers helps in determining these parameters precisely.

Whole Numbers

Whole numbers are essentially the non-negative integers. This means they encompass all the natural numbers including zero. Let's dive into what makes whole numbers special:

• Whole numbers include numbers like 0, 1, 2, 3, and so forth. Importantly, they do not include negative numbers.
• In contexts like counting or measuring quantity, whole numbers are used because they can represent complete items or units.
• Whole numbers do not have fractions or decimals. Each is a distinct, complete number.

In the world of inequalities, sometimes problems may refer to whole numbers when they mean integers that could be either positive or negative. This can be confusing, as it was in the original exercise. It's crucial to read mathematical problems carefully to understand what is specifically being asked. In this exercise, the term 'whole number' was interpreted to mean all integers fulfilling the inequality \(-8 < x \leq -2\). This distinction is significant in communicating correctly what values can replace \(x\).

Negative Numbers

Negative numbers are numbers less than zero. They play an important role in mathematics, especially when dealing with inequalities and ranges:

• Negative numbers are used to represent values that are less than zero, such as debts or temperatures below freezing.
• In an inequality, negative numbers affect the direction: for \(-8 < x \leq -2\), the solution considers all negative integers up to and including negative two.
• Understanding negative numbers is essential for solving problems across algebra, calculus, and everyday scenarios like finance.

In the particular exercise, negative numbers are crucial as \(x\) represents values that are between \(-8\) and \(-2\), inclusive, on the negative number line. This illustrates how negative values integrate seamlessly with other integer types to represent a vast range of possible numbers within a given set of conditions.