Problem 49 - 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. $$5 x+5=5(x+3)-3$$

Step-by-Step Solution

Answer

The equation is a contradiction with no solution.

Step 1: Identify the Equation

We start with the given equation: \[ 5x + 5 = 5(x + 3) - 3 \]

Step 2: Simplify Both Sides

Distribute the terms on the right side of the equation:\[5x + 5 = 5x + 15 - 3\]Simplify the expression further:\[5x + 5 = 5x + 12\]

Step 3: Isolate Variable Terms

Next, subtract \(5x\) from both sides of the equation to eliminate the \(x\) term:\[5 = 12\]

Step 4: Analyze the Result

The resulting equation, \(5 = 12\), is false. Since there are no variables left and the equation is untrue, we have a contradiction.

Step 5: Conclusion on Type of Equation

Since simplifying the equation resulted in a false statement, without any true values of \(x\), this equation is a contradiction. It has an empty solution set.

Key Concepts

Contradiction Equation

A contradiction equation is one where there is no possible value for the variable that will satisfy the equation. This means that, after simplifying, you end up with a statement that is false and cannot be true under any circumstances. For instance, the equation \(5x + 5 = 5x + 12\) simplifies to \(5 = 12\), which is a false statement. With no valid solution, we say that this equation is a contradiction. Such equations have an empty solution set, meaning there is no number that can be substituted for \(x\) to make the equation true. Recognizing a contradiction involves simplifying the equation until a false statement surfaces, indicating an inherent conflict in the equation itself.

Identity Equation

An identity equation is an equation that is always true, no matter what value is substituted for the variable. This happens when, after simplifying, the left and right sides of the equation are the same for all values of the variable. For example, if you have \(2x + 3 = 2x + 3\), this simplifies to \(0 = 0\), a true statement regardless of the value of \(x\). Identity equations are unique because they indicate that the equation holds universally. In these cases, the solution set includes all possible values, since any number could be plugged into the variable to satisfy the equation. Identifying identity equations helps understand which equations are universally applicable.

Conditional Equation

Conditional equations are equations that hold true only for specific values of the variable. Unlike identity equations, conditional equations simplify to reveal certain values that make them true. For example, consider the equation \(x - 2 = 3\), which simplifies to \(x = 5\). Here, only when \(x\) equals 5 is the equation true. Conditional equations typically result in a non-empty, but not universal, solution set, meaning only specific values satisfy them. Comprehending conditional equations is essential for problem-solving as it allows us to pinpoint exact solutions and understand the conditions under which an equation remains valid.

Solution Set Analysis

Solution set analysis refers to examining an equation to determine the set of all possible values that satisfy the equation. It involves identifying whether the solution set is empty, contains specific solutions, or includes all values.
This analysis is crucial because:

• It tells you if you have a contradiction, identity, or conditional equation.
• It helps in comprehending the nature of the equation and its results.
• It guides you in predicting outcomes for real-world problems modeled by equations.

For the equation \(5x + 5 = 5x + 12\), the solution set is empty, indicating a contradiction. Solution set analysis is a fundamental skill for interpreting equations and understanding mathematical relationships.