Problem 46 - 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 \(x\) more than 2 units from 0

Step-by-Step Solution

Answer

The absolute value inequality is \(|x| > 2\).

Step 1: Understand the Phrase

The phrase 'all real numbers \(x\) more than 2 units from 0' means we want to include numbers on both sides of 0 that are at least 2 units away. So, \(x\) can be either greater than 2 or less than -2.

Step 2: Set Up Absolute Value Inequality

To express this as an inequality involving absolute value, recall that \(|x|\) measures the distance of \(x\) from 0. We want numbers that are more than 2 units away from 0, which means we need \(|x| > 2\).

Key Concepts

Real Numbers

Real numbers are the type of numbers we use most often in everyday life. They include all the numbers you can think of, like whole numbers, fractions, and decimals. Real numbers can be positive, negative, or even zero, forming a continuum that includes the smallest to the largest possible numbers on a number line. When we talk about real numbers in mathematics, we refer to values that can represent a distance, such as inches, miles, and even abstract distances like that in an equation.

• All counting numbers (1, 2, 3, etc.) are real.
• Fractions such as \( \frac{1}{2} \) or decimals like 3.14 are real numbers.
• Negative numbers, like -5, also fall in this category.

Understanding real numbers helps solve different problems since they provide an entire spectrum to define where a number lies and how it compares to others. They are the building blocks for more advanced math topics.

Inequalities

In mathematics, inequalities describe the relative size or order of two values. They tell us whether one number is greater than, less than, or possibly equal to another number. Inequalities are often used when solving problems where the exact number isn't known but a range is provided instead. Here are some symbols you'll often encounter:

• \( > \): greater than
• \( < \): less than
• \( \geq \): greater than or equal to
• \( \leq \): less than or equal to

The expression of inequalities can be visualized on the number line by marking the solution set and shading the appropriate area. This provides a visual representation of where values that satisfy the inequalities would lie. Inequalities allow us to use a range of numbers as solutions rather than pinpointing a specific number.

Distance Concept

The distance concept in mathematics is closely related to absolute value, which represents the magnitude or the "how far" aspect of a number from zero, disregarding its direction. When considering the expression of distance, the absolute value \( |x| \) signifies how many units a number is away from the origin point of zero on a number line, always as a positive measure.For instance:

• The absolute value of 3 is \( |3| = 3 \).
• Similarly, \( |-3| = 3 \), since -3 is also three units away from zero.

In the context of inequalities, understanding this "distance" helps set conditions for numbers that meet specific criteria, such as all numbers more than 2 units away from zero need their absolute value to be greater than 2. In mathematical terms, the inequality \( |x| > 2 \) describes all the real numbers either less than -2 or greater than 2.