Problem 28 - Beginning and Intermediate Algebra

Question

Give three numbers between -6 and 6 that satisfy each given condition. Rational numbers but not negative numbers

Step-by-Step Solution

Answer

0, 1, 2

Step 1: Understand the Problem

Identify three rational numbers between -6 and 6 that are not negative.

Step 2: Identify Rational Numbers

Rational numbers are numbers that can be expressed as a fraction of two integers, where the denominator is not zero. For this problem, we can consider any fraction or whole number within the given range.

Step 3: Avoid Negative Numbers

Since the solution requires numbers that are not negative, consider only numbers that are zero or positive.

Step 4: Select Appropriate Numbers

Choose any three rational numbers in the range (-6, 6) that are non-negative. Examples include 0, 1, and 2. These numbers are rational and lie within the specified range.

Step 5: Verify the Solution

Check that the chosen numbers (0, 1, 2) are within the range and meet the condition of being rational and non-negative.

Key Concepts

identifying rational numbers

Rational numbers are essentially numbers that can be expressed as a fraction \(\frac{a}{b}\) where both \(a\) and \(b\) are integers and \(b\) is not zero. This means that when you see a number like 1.5, it can also be written as \(\frac{3}{2}\), making it a rational number. Here are more examples of rational numbers you frequently encounter:

• \(\frac{1}{2}\)
• \(-\frac{3}{4}\)
• Any whole number, like 5 or -7, since they can be expressed as fractions like \( \frac{5}{1} \) or \( \frac{-7}{1} \).

Remember, as long as you can write the number as a fraction with integers and the denominator is not zero, it qualifies as rational. Irrational numbers, like \( \sqrt{2} \) or \(\frac{\text{pi}}{1}\), cannot be represented this way.

range of numbers

When dealing with math problems, understanding the 'range' is crucial. In this exercise, the range of numbers given is between -6 and 6. This means we are looking at all numbers that fall from -6 to 6, excluding the endpoints unless specified otherwise. Considering our task to identify rational numbers within this range, we need numbers that:

• Are greater than -6
• Are less than 6

This range includes every possible number that fits within these boundaries, including fractions and whole numbers. So for instance, 0, 2.5, and -5.9 fall within this range, while -6.1 and 6.1 do not. Think of the range as the boundaries of your playground—any valid number within these walls is fair game for our solution.

non-negative numbers

Non-negative numbers are numbers that are either positive or zero. Essentially, they are never less than zero. In mathematical notation, these numbers are denoted as \( x \geq 0\). Examples include:

• Whole numbers: 0, 1, 2, etc.
• Positive fractions: \(\frac{1}{2}\), \(\frac{3}{4}\) , etc.

It's key to remember that non-negative numbers don't include anything negative. If we only consider rational numbers (as we learned previously), then any number that is rational and greater than or equal to zero fits this category.
In our exercise, we purposefully steer clear of any negatives, focusing on picking numbers such as 0, 1, and 2, ensuring they fit both non-negative and rational criteria.