Problem 48 - 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 \(x\) at most 4 units from 2

Step-by-Step Solution

Answer

\(|x - 2| \leq 4\)

Step 1: Understand the Phrase

The phrase "all real numbers \(x\) at most 4 units from 2" refers to the numbers that have a distance of 4 or less from the number 2 on the number line.

Step 2: Identify the Center and Range

The center of our set is 2, and the maximum distance from this center is 4 units. This means any number within 4 units of 2, both to the left and right on the number line, should be included.

Step 3: Use Absolute Value to Express Distance

The absolute value \(|x - 2|\) represents the distance of \(x\) from 2. We need this distance to be at most 4 units. Thus, we set up the inequality: \[ |x - 2| \leq 4 \]

Step 4: Finalize the Inequality

The inequality \(|x - 2| \leq 4\) encapsulates all real numbers \(x\) that are 4 units or less from 2. This includes all \(x\) between \(-2\) and \(6\) inclusive.

Key Concepts

Real Numbers

Real numbers are the set of all numbers that can be represented on the number line. This includes both rational numbers, like fractions and integers, and irrational numbers which cannot be exactly expressed as fractions such as \( \sqrt{2}\) or \( \pi \). Real numbers are essential in representing real-world quantities. Some properties of real numbers include:

• Completeness: Between any two real numbers, no matter how close, there exists another real number.
• Additive Identity: Adding 0 to a real number leaves it unchanged.

Real numbers can be positive, negative, or zero, and help us measure distances, lengths, and even abstract concepts like probabilities.

Distance on Number Line

Distance on a number line is a measure of how far apart two numbers are. It is always a positive value because distance itself cannot be negative. In the context of absolute value inequalities, the distance is represented using absolute values. The absolute value of a number, \(|x|\), is the number without regard to its sign. Thus, the absolute value of the difference between two numbers gives the distance on a number line.For instance, in the inequality \(|x - 2| \leq 4\), the expression \(|x - 2|\) represents the distance of any real number \(x\) from 2. We interpret this to find all numbers whose distance from 2 is 4 or less. These numbers lie between -2 and 6 on the number line.

Inequality Expression

Inequality expressions are used to denote the relationship between two values or expressions that are not equal but related in some way, either by being greater than, less than, or possibly equal to, a given value. They are essential in areas like algebra, calculus, and real-world problem-solving.An inequality involving absolute value helps express a range of values. For example, \(|x - 2| \leq 4\) is an inequality that includes all real numbers \(x\) with a distance of at most 4 from 2. This can be visualized on a number line as including all values from -2 to 6.Some common absolute value inequalities are:

• \