Problem 48 - 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. $$ 0.5(3 x-1)+0.5 x=2 x-0.5 $$

Step-by-Step Solution

Answer

The equation is an identity.

Step 1: Expand the Equation

Let's start by expanding the equation on both sides. The given equation is \(0.5(3x - 1) + 0.5x = 2x - 0.5\). Distribute the 0.5 across the terms in the parentheses: \(0.5 \cdot 3x - 0.5 \cdot 1 + 0.5x = 2x - 0.5\). This simplifies to \(1.5x - 0.5 + 0.5x = 2x - 0.5\).

Step 2: Combine Like Terms on the Left Side

Combine the terms on the left side of the equation. You have \(1.5x + 0.5x\), which equals \(2x\). The equation becomes \(2x - 0.5 = 2x - 0.5\).

Step 3: Simplify the Equation

The equation is now \(2x - 0.5 = 2x - 0.5\). Observe that if we subtract \(2x\) from both sides, we are left with \(-0.5 = -0.5\).

Step 4: Determine the Nature of the Equation

Since the simplification \(-0.5 = -0.5\) is always true, the original equation holds for all values of \(x\). Therefore, the equation is an identity.

Key Concepts

Contradiction in Equations

Sometimes, when solving an equation, you might end up with a statement that doesn't make sense. This is called a contradiction. If you ever see something like \( 5 = 3 \) after simplifying your equation, you've found a contradiction.
This means that no possible value of the variable can make the equation true.Here are the key points to remember about contradictions:

• Contradictions typically show two unequal values as equal.
• If an equation simplifies to a contradiction, it means no solution exists.
• In equations, contradictions reveal inconsistencies in assumptions or errors.

Always check your work if you find a contradiction; it might mean there's a mistake, or it could simply mean the equation truly has no solution.

Understanding Identity Equations

An identity is like a magical equation that is always true, no matter what value is plugged into the variable. In these cases, you'll finish simplifying the equation and end up with something eternally true, like \( 0 = 0 \) or \( 3x = 3x \).Here’s why identity equations are special:

• The left side and the right side of the equation are exactly the same after simplification.
• Identity equations have an infinite number of solutions because they hold true for any variable.
• If after solving you see a valid identical statement, your equation is an identity.

In the exercise example, when we solved the equation \( 0.5(3x - 1) + 0.5x = 2x - 0.5 \) and simplified it to \( -0.5 = -0.5 \), this told us the equation is an identity, true for any \( x \).

What Makes an Equation Conditional?

Conditional equations have a special requirement: they are only true if the variable takes on a specific value. Imagine there’s a secret key that unlocks the truth of the equation, and that key is the variable's exact value.Here's what characterizes a conditional equation:

• If an equation reduces to a statement like \( x = 4 \), it's conditional because only \( x = 4 \) fulfills it.
• Conditional equations have exactly one solution, unlike identities or contradictions.
• The solution tells precisely which value makes both sides of the equation equal.

If during your equation-solving journey you find a specific value for \( x \) that works, you've got a conditional equation. While this isn't the case with our exercise example, it's quite different from identities and contradictions.