Problem 50 - 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. $$5-4 x=5 x-(9+9 x)$$

Step-by-Step Solution

Answer

The equation is a contradiction with an empty solution set \(\emptyset\).

Step 1: Simplify Both Sides of the Equation

Begin by distributing the terms on the right-hand side of the equation: \[ 5 - 4x = 5x - 9 - 9x \]Simplify the equation by combining like terms. On the left side, it's already simplified: \[ 5 - 4x \]On the right side, combine \(5x - 9x\) to get \(-4x\):\[ 5x - 9 - 9x = -4x - 9 \] So the equation is now: \[ 5 - 4x = -4x - 9 \]

Step 2: Eliminate Like Terms

Move one of the \(-4x\) terms to the other side by adding \(4x\) to both sides of the equation:\[ 5 - 4x + 4x = -4x + 4x - 9 \]This simplifies to:\[ 5 = -9 \]

Step 3: Analyze the Simplified Equation

Observe the resulting equation: \[ 5 = -9 \]This statement is false for any value of \(x\), meaning there are no solutions that make the equation true.

Step 4: Determine the Type of Equation

Since the simplified form \( 5 = -9 \) is always false, classify the equation as a contradiction. Contradiction means there is no solution that satisfies the equation.

Step 5: Conclusion on the Solution Set

Since there are no solutions, the solution set is the empty set, denoted by \( \emptyset \).

Key Concepts

Understanding Contradictions in Equations

An equation is classified as a contradiction when it is impossible for any value of the variable to make the equation true. This means that no real number will satisfy the equation. A contradiction results in a statement that is always false, such as \( 5 = -9 \). When simplified, these equations do not equate logically, proving their contradictory nature.

Here's how contradictions are handled:

• Simplify both sides of the equation as much as possible.
• Look for a resulting statement that is always false, independent of the variable.
• If such a false statement exists, the equation is a contradiction.
• The solution set for a contradiction is the empty set, denoted by \( \emptyset \), because there are no possible solutions.

Being able to identify a contradiction helps in solving more complex algebraic problems by narrowing down the possibilities early on.

The Nature of Identity Equations

Identity equations are special because they are always true, no matter what value the variable takes. Unlike contradictions, identity equations mean that both sides of the equation are equivalent expressions. For example, an equation like \( x + 2 = x + 2 \) is an identity since any value of \( x \) will satisfy the equation.

To identify an identity equation, you should:

• Simplify both sides of the equation completely.
• Ensure the remaining expressions are equivalent throughout, independent of variables.
• If equivalent, the equation is recognized as an identity.
• The solution set encompasses all real numbers, showcased by \( (-\infty, \infty) \).

Understanding identities is crucial, as they reinforce the idea of equivalent forms and allow for simplifications in algebraic expressions.

Understanding Conditional Equations

Conditional equations are true for specific values of the variable involved. They are unlike contradictions, which are never true, and identities, which are always true. A conditional equation will yield a finite, specific solution set. For example, consider the equation \( 3x + 1 = 10 \), which is only true for \( x = 3 \).

To recognize a conditional equation, follow these steps:

• Simplify both sides of the equation as much as possible.
• Isolate the variable to find its specific value(s).
• If a specific, finite solution exists, it is a conditional equation.
• The solution set reflects the specific values that satisfy the equation, like \( \{x = 3\} \).

Conditional equations are very common in algebra, allowing for the identification of precise solutions in mathematical and real-world problems.