Problem 44 - College Algebra with Modeling and Visualization

Question

Complete the following. (a) Solve the equation symbolically. (b) Classify the equation as a contradiction, an identity, or a conditional equation. $$ \frac{1}{2} x-2(x-1)=-\frac{3}{2} x+2 $$

Step-by-Step Solution

Answer

The equation is an identity because it holds true for all values of \(x\).

Step 1: Distribute the Terms

Let's start by distributing the terms within the parentheses on the left side of the equation. Distribute \ -2 to \ (x - 1) to get: \\( \frac{1}{2} x - 2(x - 1) = -2x + 2 \). This simplifies to: \\( \frac{1}{2} x - 2x + 2 = -\frac{3}{2} x + 2 \).

Step 2: Combine Like Terms

On the left side of the equation, combine the like terms: - \( \frac{1}{2} x \) and \( -2x \) combine to form: \(-\frac{3}{2}x\). So the equation becomes:\(-\frac{3}{2}x + 2 = -\frac{3}{2}x + 2\).

Step 3: Analyze the Equation

Notice that both sides of the equation are exactly the same: \(-\frac{3}{2} x + 2 = -\frac{3}{2} x + 2\).This observation is important for classifying the type of equation.

Step 4: Classify the Equation

Since both sides of the equation are equal for all values of \(x\), we classify this as an identity equation. An identity equation means that any real number substituted for \(x\) will satisfy the equation.

Key Concepts

Conditional Equation

A conditional equation is one that is true only for certain values of the variable involved. In simpler terms, it depends on the condition of the variable to hold true. For example, the equation \( x + 3 = 5 \) is only true when \( x \) is equal to 2. For any other value of \( x \), the equation does not hold true.

Key points about conditional equations include:

• They often arise in solving algebraic problems where solutions are constrained to certain values.
• They indicate specific points on a graph, like intersections or certain regions.
• Solving these involves isolation of the variable to find these specific solutions.

Recognizing a conditional equation involves checking if it is satisfied by only particular values of the variable rather than a broad range or all values.

Identity Equation

An identity equation is particularly interesting because it is true for every possible value of the variable. This kind of equation usually represents a universal truth in algebra. In the step-by-step solution, we found that both sides of the equation are exactly the same: \(-\frac{3}{2} x + 2 = -\frac{3}{2} x + 2\). This is a classic example of an identity equation.

Why are identity equations significant?

• They represent an infinite number of solutions – any value substituted in place of \( x \) will satisfy the equation.
• They often reaffirm an algebraic simplification or property rather than providing a specific numerical solution.
• Understanding them helps in discerning whether an algebraic operation yields an equivalent expression.

Identity equations are crucial in checking equivalency across different mathematical expressions.

Contradiction Equation

A contradiction equation is one that has no possible real solutions because it is always false. Think of it as a mathematical paradox where no real value of the variable will satisfy the equation. For instance, the equation \( x + 2 = x - 3 \) cannot be true for any real number because adding 2 to \( x \) will never equal \( x \) minus 3.

Characteristics of contradiction equations include:

• They result in statements that are impossible, like \( 0 = 5 \), after simplifying the original equation.
• They can indicate that an assumed relationship or condition is incorrect.
• These equations play a key role in logical reasoning and determining bounds or limitations in mathematical problems.

Contradiction equations, although resulting in no solutions, help define the boundaries between viable and non-viable mathematical relationships.