Problem 30 - Fundamentals of Mathematics

Question

For the following 5 problems, what numbers can replace \(m\) so that the following statements are true? \(-5 \leq m<5, m\) a whole number.

Step-by-Step Solution

Answer

The values of \(m\) are -5, -4, -3, -2, -1, 0, 1, 2, 3, 4.

Step 1: Understanding Inequality

The inequality given is \(-5 \leq m < 5\) which means that \(m\) is greater than or equal to -5 and less than 5. Keep in mind that \(m\) should be a whole number. Whole numbers are non-negative integers like 0, 1, 2, etc.

Step 2: Identify the Range for Whole Numbers

Since \(-5 \leq m < 5\), we need to identify all integers in this range. Whole numbers, by their regular definition, comprise non-negative integers, but the given problem includes negative numbers using this term.

Step 3: List Possible Values of m

Given the understanding of the interval and the inclusion of negative whole numbers, the values for \(m\) span from -5 to 4, because the inequality is exclusive on the right side. Therefore, whole numbers in this context are: \[-5, -4, -3, -2, -1, 0, 1, 2, 3, 4\]. These are all the integer values satisfying the condition.

Key Concepts

Whole Numbers

Whole numbers are a special type of numbers that include all non-negative integers. In everyday math, when we mention whole numbers, we primarily consider numbers that start from 0 and go upwards, such as 0, 1, 2, 3, and so on.

• Whole numbers do not include negative numbers or fractions.
• They also do not include decimal numbers; each number is whole in itself, consisting of complete units.

However, sometimes in mathematical problems, like this exercise, the term "whole numbers" might be used alongside negative numbers. It's essential to understand the context correctly.
In the context of this inequality problem, when they say whole numbers, they are including both negative integers and zero. Therefore, the numbers that meet the criteria of being whole numbers in the given inequality span both negative and positive numbers.

Integer Ranges

An integer range in math specifies a span of whole numbers from one point to another. In inequalities, these ranges are often defined by using symbols like \( \leq \) (less than or equal to) and \(<\) (less than).
For example, the inequality \(-5 \leq m < 5\) defines a range of integers that starts from \(-5\) and goes up to but does not include \(5\).

Important points regarding integer ranges:

• The symbol \(\leq\) indicates that the endpoint is included in the range.
• The symbol \(<\) means the endpoint isn't included in the range, as is the case with 5 in our problem.

Integer ranges allow us to effectively understand what numbers are applicable within a certain set or condition. It enables identification of all possible values that satisfy an inequality, which is crucial for solving problems like this one, where understanding the specific numbers that \(m\) can take is necessary.

Solving Inequalities

Solving inequalities involves finding all the possible values of a variable that satisfy the given inequality. It is a step-wise logical process where we deduce a solution set based on the conditions provided.
For example, the problem we're facing is determining the values of \(m\) that fulfill the inequality \(-5 \leq m < 5\). Here's a simple strategy to solve such inequalities:

• Read the Inequality: Understand what the inequality symbol means. Here, \(\leq\) and \(<\) show included and excluded boundaries respectively.
• Identify Possible Values: List all whole numbers within the range from \(-5\) (inclusive) to 5 (exclusive).
• Verify the Values: Check each number to ensure it satisfies the inequality. Each value from \(-5\) up to \(4\) meets the condition given.

By systematically approaching inequalities, you can ensure you capture all potential solutions correctly. These skills are useful in every branch of mathematics, as they handle ranges, limits, and finding applicable values within constraints.