Problem 72 - Algebra for College Students

Question

Use an algebraic approach to solve each problem. Make up an equation whose solution set is the set of all real numbers and explain why this is the solution set.

Step-by-Step Solution

Answer

An example of such an equation is \( x = x \); it holds for any real number \( x \).

Step 1: Understand the Problem

To create an equation whose solution set is the set of all real numbers, the equation must hold true for any possible value of the variable.

Step 2: Choose a Suitable Equation Form

A simple form of equation that holds true for all values of the variable is an identity. An identity is an equation that is true for all values of the variable.

Step 3: Construct the Equation

An example of such an equation is \( x = x \). This equation is satisfied for any real number substituted for \( x \) since both sides of the equation will always be equal.

Step 4: Validate the Equation

Substitute any real number into the equation \( x = x \). For instance, \( x = 2 \) gives \( 2 = 2 \), \( x = -5 \) gives \( -5 = -5 \), and \( x = 0 \) gives \( 0 = 0 \). In all cases, the left side equals the right side.

Step 5: Reasoning the Solution Set

Since the equation holds for any substitution of \( x \), this means the set of all solutions is the set of all real numbers. Thus, all real numbers make the equation true.

Key Concepts

Solution Set

When we talk about the **solution set** of an equation, we are referring to all the values of the variable that satisfy the equation. Essentially, it's the collection of answers or solutions that make the equation true.

In the context of the given exercise, the task was to find an equation whose solution set includes all real numbers. This means any real number can be substituted for the variable in the equation, and the result will still hold true. The solution set is important because it tells us which values satisfy a given algebraic statement.

For example, in the equation \( x = x \), every real number is a part of the solution set because substituting any real number for \( x \) makes both sides of the equation equal, thus satisfying the equation.

Real Numbers

Real numbers are a fundamental concept in mathematics. They include all the numbers that can be found on the number line. Real numbers encompass a wide range: decimals, fractions, integers, and whole numbers.

In this exercise, when we say the solution set is the set of all real numbers, we're indicating any number you can think of—positive or negative, big or small, whole or fractional—can be a solution. This is a broad solution set, meaning there are infinite possibilities.

• Integers like -3, 0, 2
• Decimals such as 0.5 and -2.75
• Fractions like \( \frac{1}{2} \) and \(-\frac{3}{4} \)

Understanding that real numbers include these types and more helps give context to why the equation \( x = x \) fits every possible real number. Each of these numbers can be plugged into the equation and make it valid.

Identity Equations

Identity equations are special types of equations in algebra. They are characterized by the fact that they are always true, no matter what value you choose for the variable.

Consider the equation \( x = x \). This is an example of an identity equation because it holds true for any substitutable value of \( x \). No matter the real number, plugging it into the equation results in equality on both sides

Identity equations are unique because they describe universal properties that generally hold true in mathematics. Unlike conditional equations which are true for certain values, identity equations embrace every possible real number, reflecting the broadest solution set possible.

These equations help illustrate theoretical concepts and offer insights into the properties and behaviors of functions and operations across all real numbers.