Problem 62 - A Graphical Approach to Precalculus with Limits

Question

Classify each equation as a contradiction, an identity, or a conditional equation. Give the solution set. Use a graph or table to support your answer. $$\frac{2 x-1}{3}=\frac{2 x+1}{3}$$

Step-by-Step Solution

Answer

The equation is a contradiction with an empty solution set.

Step 1: Identify and Simplify the Equation

We are given the equation \( \frac{2x - 1}{3} = \frac{2x + 1}{3} \). Since both sides of the equation have the same denominator, we can multiply both sides by 3 to eliminate the fractions, simplifying the equation to \( 2x - 1 = 2x + 1 \).

Step 2: Isolate x

Next, we'll try to isolate \(x\) by subtracting \(2x\) from both sides of the equation: \(2x - 2x - 1 = 2x - 2x + 1\), which simplifies to \(-1 = 1\).

Step 3: Classify the Equation

The equation \(-1 = 1\) is a contradiction because it is a false statement. This means there is no possible value of \(x\) that can satisfy the equation. Therefore, the solution set is empty.

Key Concepts

Understanding Contradictions in Equations

In mathematics, a contradiction occurs when an equation simplifies to an untrue statement. Unlike the usual arithmetic where equations are about finding values that make a sentence true, a contradiction is when the math tells us there’s no possible solution. For example, consider the equation \[-1 = 1.\] Clearly, this statement is false because negative one cannot equal positive one. When simplifying equations, as with our example equation given \[\frac{2x-1}{3} = \frac{2x+1}{3},\]you subtract terms and eventually end up with a false statement. This indicates that no number can satisfy the condition set by the equation, thus, we call this a contradiction. When faced with a contradiction, remember, the equation simply has no solution.

The Notion of a Solution Set

In mathematics, a solution set is the collection of all possible values that satisfy an equation. It tells us which numbers, if any, solve the equation. For equations that are contradictions, just like in our case of \[\frac{2x-1}{3} = \frac{2x+1}{3},\]after simplifying we get \[-1 = 1,\] suggesting no solution exists. This leads us to an empty solution set, denoted by \[\emptyset.\] In other words, there are no values of \(x\) that can make the equation true. In broader terms:

• If an equation is always true, regardless of the value of \(x\), the solution set contains every number.
• If specific numbers satisfy the equation, those numbers form the solution set.
• In contradictions, however, the solution set remains empty.

Understanding what a solution set is allows you to classify equations more effectively.

Simplifying Equations: The Basics

Simplifying equations is a fundamental step in solving mathematical problems. It's like cleaning up the clutter to see what's really happening. In essence, we perform operations to make the equation more straightforward or even solve it. Let's break it down:

• **Remove fractions**: Multiply both sides by the least common multiple to eliminate denominators.
• **Collect like terms**: Bring similar terms together, such as adding all \(x\) terms and constant terms separately to one side.
• **Isolate variables**: Get the variable \(x\), or any variable of interest, by itself on one side of the equation.

For instance, with the equation \[\frac{2x - 1}{3} = \frac{2x + 1}{3},\] by multiplying both sides by 3, the equation becomes simpler as the fraction is removed: \[2x - 1 = 2x + 1.\] Then by subtracting \(2x\) from both sides, we end up simplifying it to \[-1 = 1.\] Although this leads to a contradiction in our case, the process of simplifying allows us to see the equation's nature more clearly, helping us to correctly identify it as having no solution.