Problem 50 - Algebra and Trigonometry

Question

\(49-52=\) A phrase that describes 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 inequality is \(|x| > 2\).

Step 1: Understanding the Phrase

The phrase "all real numbers \(x\) more than 2 units from 0" means we are looking for values of \(x\) that are either more than 2 units to the right of 0 or more than 2 units to the left of 0.

Step 2: Defining More Than 2 Units from 0

A distance of more than 2 units from 0 encompasses values that are either greater than 2 or less than -2. This can be expressed using two inequalities: \(x > 2\) or \(x < -2\).

Step 3: Using Absolute Value

The absolute value \(|x|\) represents the distance of \(x\) from 0 on the number line. For \(x\) to be more than 2 units away from 0, the absolute value must be greater than 2.

Step 4: Formulating the Absolute Value Inequality

To express the idea of more than 2 units away from 0 using an inequality with absolute value, we write: \(|x| > 2\). This captures both conditions \(x > 2\) and \(x < -2\) in one single inequality.

Key Concepts

Real Numbers: The Building Blocks of Math

Real numbers include all the numbers we encounter in everyday life. They consist of both rational numbers, such as integers and fractions, and irrational numbers, which cannot be expressed as simple fractions, like \( \sqrt{2} \). Real numbers can be thought of as the points on an infinitely long number line.
This comprehensive set of numbers is crucial because it includes both positive and negative numbers, along with zero. They form the backbone of most mathematical calculations involving continuous quantities. When solving any inequality, particularly ones involving absolute values, understanding the nature of real numbers is essential. We need to appreciate this wide set in order to handle the different outcomes or solutions that these inequalities might present.

Inequalities: Comparing Values

Inequalities are mathematical expressions used to compare two values. Unlike equations, which show equality, inequalities demonstrate a range of possible solutions. They involve symbols like \( >, <, \ge, \le \), which respectively mean greater than, less than, greater than or equal to, and less than or equal to.
Inequalities arise when defining conditions or constraints in problems. For instance, in our exercise, we're told that real numbers are more than a certain distance from zero. Here, inequalities \( x > 2 \) or \( x < -2 \) quantify how numbers lie on the number line relative to the origin.
Understanding inequalities allows us to express conditions in a flexible way, opening up refined approaches to solving problems in algebra and calculus.

Absolute Value Expressions: Measuring Distance on the Number Line

An absolute value expression measures how far a number is from zero on the number line, regardless of direction. It is denoted by two vertical bars, for example, \( |x| \). The absolute value of any real number is always non-negative, treating both sides of the origin uniformly.
To express a condition like "more than 2 units from 0", we use an absolute value inequality, \( |x| > 2 \.\) This not only indicates numbers more than 2 steps away from zero but elegantly captures both extremes, \( x > 2 \) and \( x < -2 \,\) in a single form.
Absolute value expressions simplify complex conditions into compact mathematical statements, ideal for scenarios involving distance and deviation in mathematics.