Problem 45 - Algebra and Trigonometry

Question

A phrase describing a set of real numbers is given. Express the phrase as an inequality involving an absolute value. All real numbers \(x\) less than 3 units from 0

Step-by-Step Solution

Answer

The inequality is \(|x| < 3\).

Step 1: Understanding the Phrase

The phrase states 'all real numbers x less than 3 units from 0.' This means we are looking for numbers that are within a distance of 3 from 0 on the number line.

Step 2: Defining Distance Using Absolute Value

The distance between a number x and 0 can be represented using absolute value as \(|x|\). Hence, if a number is less than 3 units from 0, it means that its absolute value is less than 3.

Step 3: Formulating the Inequality

Based on our understanding, the inequality can be expressed using absolute value as \(|x| < 3\). This means x can be anywhere in the range where the distance from 0 is less than 3.

Key Concepts

Understanding Real Numbers

Real numbers form the backbone of mathematics—they include all the numbers you encounter in daily life and more. Real numbers cover both rational numbers, like 7, -2, and 1/2, and irrational numbers, such as \(\sqrt{2}\) and \(\pi\). Every point on the number line corresponds to a real number, making the number line a practical tool for visualizing these concepts.
Real numbers can be positive, zero, or negative. Their complete nature means they can represent decimal expansions, whether the decimals are repeating or non-repeating. This versatility is why they are crucial in solving equations, understanding limits, and analyzing growth patterns. In the context of our exercise, knowing that we are dealing with real numbers means we are examining every possible numerical value along a continuum without restriction to only integers or rational numbers.

Distance On the Number Line

The number line is a simple yet powerful visual tool to represent numbers and understand concepts like distance. To picture this, imagine a line stretching infinitely in both directions with zero in the center. Numbers to the right are positive, and those to the left are negative.
Distance on the number line is a measure of how far apart two numbers are, regardless of direction. This is where absolute value comes into play, denoting the distance of a number from zero. So, \(|x|\) indicates the distance from 0 to a number \(x\). For example, both 5 and -5 are 5 units away from zero. Thus, the absolute value focuses purely on magnitude.
In the given exercise, the phrase "less than 3 units from 0" suggests that the distance of \(x\) from the origin must be under 3 units no matter the direction on the number line. Hence, \(|x|<3\) effectively captures this range.

Solving Absolute Value Inequalities

Inequalities help us describe conditions where one value is smaller or larger than another. As in many cases related to distance, inequalities using absolute values are very powerful. They describe the range within which real numbers fall.
An inequality like \(|x| < 3\) tells us that the real number \(x\) is any number from -3 to 3, as the absolute value of \(x\) is less than 3. The solutions to such an inequality include all real numbers that satisfy this condition. This method provides a concise way to express complex ideas such as constraints on distances.
Solving these involves understanding that an inequality like \(|x| < a\) means \(-a < x < a\). In contrast, \(|x| > a\) would mean \(x < -a\) or \(x > a\). It's essential to know that solving inequalities requires ensuring that both sides preserve the inequality property, so multiplying or dividing by negative numbers flips the inequality sign. Understanding these steps is crucial to solving absolute value inequalities accurately.