Problem 62 - A Graphical Approach to College Algebra

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 and has an empty solution set, \( \emptyset \).

Step 1: Identify the Equation Type

First, let's analyze the given equation \( \frac{2x-1}{3} = \frac{2x+1}{3} \). Notice that both sides of the equation have the same denominator (3). This suggests that if the numerators are not equal, the equation might be based on specific values of \( x \).

Step 2: Eliminate the Common Denominator

To simplify the equation, multiply each term by 3 to eliminate the denominator:\[ 3 \times \frac{2x-1}{3} = 3 \times \frac{2x+1}{3} \]This reduces to:\[ 2x - 1 = 2x + 1 \]

Step 3: Simplify and Analyze the Result

Subtract \( 2x \) from both sides to isolate terms without \( x \):\[ 2x - 1 - 2x = 2x + 1 - 2x \]This simplifies to:\[ -1 = 1 \]This statement is not true, thus it shows that there are no values of \( x \) that can satisfy the equation.

Step 4: Classify the Equation

Since the simplified equation \( -1 = 1 \) is not true, and it never holds for any value of \( x \), this equation is a contradiction. A contradiction is an equation that has no solution.

Step 5: Determine the Solution Set

Since the equation is a contradiction, the solution set is empty. In set notation, this is represented as \( \emptyset \).

Key Concepts

Solution Set

A solution set is a collection of all values that satisfy a given equation. For most equations, we'll find one of three types of solution sets: a single solution, multiple solutions, or no solutions at all. In the world of equations, the solution set allows us to determine tangible answers or truths about the equation's behavior.

To find the solution set, you need to solve the given equation. Let's take the equation \( \frac{2x-1}{3} = \frac{2x+1}{3} \). By simplifying it, we realize we reach an untrue statement, \( -1 = 1 \). Since there is no value of \( x \) that can make this statement true, the solution set is empty. This is represented in set notation as \( \emptyset \). This means no solution is found in the scope of real numbers.

Equation Analysis

Equation analysis involves breaking down the components of an equation to understand its nature. To begin, examine both sides of the equation. Identifying terms, coefficients, and constants is crucial – this helps to identify patterns or discrepancies.

Let’s apply this to the given equation: \( \frac{2x-1}{3} = \frac{2x+1}{3} \). The equation's structure reveals equal denominators, suggesting the numerators must also match for the equation to be true. By removing the denominator and simplifying: \( 2x - 1 = 2x + 1 \), it's evident through subtraction that it simplifies to \( -1 = 1 \), a false statement. Such analysis helps to classify this equation and comprehend the absence of real solutions.

Identity Equation

An identity equation is unique because every possible value for the variable will satisfy the equation, making it universally true. These equations are solidly built upon equalities that hold under all circumstances.

For example, the equation \( x = x \) is an identity, as any value substituted for \( x \) would still result in a true statement. However, our original equation was not an identity equation. When simplified, the equal sign led to a falsehood, \( -1 = 1 \), indicating that the identity feature is absent. Identity equations always possess infinite solutions, whereas our case leads to none.

Conditional Equation

A conditional equation is only true for specific values of the variable, unlike an identity equation which holds for all values. The role of a conditional equation is to express conditions or constraints through which the equation holds valid.

Consider an equation like \( x + 5 = 10 \). Here, the equation holds true when \( x = 5 \). This specific value of \( x \) is what makes the equation conditional. It resembles an agreement that only certain circumstances (or numeric conditions) fulfill.

In our initial problem, we didn't deal with a conditional equation because, post-simplification, it turned into a contradiction, unable to fulfill any condition of truth for \( x \). Conditional equations have limited, specific solutions, contrasting the undefined nature of our original equation's solution set.