Problem 49 - 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 \(x\) less than 3 units from 0

Step-by-Step Solution

Answer

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

Step 1: Understanding the Phrase

The phrase asks for all real numbers that are less than 3 units away from 0 on the number line. This means that the distance of any number \(x\) from 0 should be less than 3.

Step 2: Setting up an Absolute Value Inequality

The absolute value \(|x|\) represents the distance of \(x\) from 0. We want this distance to be less than 3, so we set up the inequality: \(|x| < 3\). This inequality captures all numbers whose distance from 0 is less than 3.

Key Concepts

Real Numbers

Real numbers are a vast collection of numbers that include a broad variety of types of numbers we use in everyday math. They encompass integers, such as -2, 0, and 5, fractions like 1/2, and irrational numbers such as \( \pi \) and \( \sqrt{2} \). This set does not include imaginary numbers. Real numbers can be placed on a number line, which means each real number corresponds to a point along this line.

• **Integers**: Whole numbers that can be positive, negative, or zero.
• **Rational numbers**: Numbers that can be expressed as the quotient of two integers.
• **Irrational numbers**: Numbers that cannot be expressed as a simple fraction.

Understanding real numbers is crucial because they are the foundation for most of the arithmetic operations and they allow us to measure and compare quantities around us.

Distance on Number Line

Distance on a number line involves understanding how far apart numbers are when plotted along a linear scale. When we talk about distance on the number line, we usually refer to the absolute value concept, which represents how far a number is from zero without considering direction - it's always a non-negative number.
For example, the distance between -3 and 0 on the number line is represented by the absolute value \(|-3| = 3\). Similarly, the distance between 3 and 0 is \(|3| = 3\). On a number line, two numbers -3 and 3 are both at equal distances from 0 but are located in opposite directions. Finding this "distance" involves subtracting the coordinates of one from the other and taking the absolute value of the result.

This concept of distance helps us solve many problems in mathematics, such as those involving absolute value inequalities, where you might need to express the solution set as the distance being less than, greater than, or equal to a certain number.

Inequality

An inequality is a mathematical statement that relates expressions that are not automatically equal. Rather than showing equal rights, like equations do, inequalities show boundaries or limits between expressions. Common symbols for inequalities include: \( < \, \leq \, > \, \geq \). These symbols help in describing the relationship between two expressions.

\(<\) or \(\leq\) indicate a number is less or less than or equal to another, whereas \(>\) or \(\geq\) imply a number is more or greater than or equal to the other. In real-world terms, it helps in determining important limits or regions on a number line where certain conditions hold true.

• **\(|x| < a\)**: The absolute value expression, where numbers have distances less than \a\ from zero.
• **\(|x| > a\)**: Represents numbers having distances greater than \a\ from zero.

Understanding inequalities is crucial because they enable us to solve and express real-world problems mathematically, wherein not only an exact solution is needed but also a range within which solutions are possible.