Problem 36 - Elementary and Intermediate Algebra

Question

Solve each equation. If an equation is an identity or a contradiction, so indicate. $$ 2(x-3)=\frac{3}{2}(x-4)+\frac{x}{2} $$

Step-by-Step Solution

Answer

It is an identity: \(-6 = -6\).

Step 1: Distribute and Simplify

First, distribute the constants on both sides of the equation. For the left-hand side: \[ 2(x-3) = 2x - 6 \]For the right-hand side: \[ \frac{3}{2}(x-4) = \frac{3}{2}x - 6 \]Thus, the equation becomes: \[ 2x - 6 = \frac{3}{2}x - 6 + \frac{x}{2} \]

Step 2: Combine Like Terms on the Right Side

Combine the terms on the right-hand side that have an 'x'. Convert them into a common denominator:\[ \frac{3}{2}x + \frac{x}{2} = \left(\frac{3}{2} + \frac{1}{2}\right)x = \frac{4}{2}x = 2x \]The equation now reads:\[ 2x - 6 = 2x - 6 \]

Step 3: Analyze the Equation

Subtract \(2x\) from both sides:\[ 2x - 6 - 2x = 2x - 6 - 2x \]Which simplifies to:\[ -6 = -6 \]Since both sides of the equation are equal, this indicates that any value for \(x\) will satisfy the equation, making it an identity.

Key Concepts

Distributive Property

When solving equations, the distributive property is one of our most important tools. It allows us to multiply a single term by each term inside a parenthesis. In our exercise, the equation starts with two parts that require the distributive property:

• On the left-hand side: \(2(x-3)\) expands to \(2x - 6\).
• On the right-hand side: \(\frac{3}{2}(x-4)\) expands to \(\frac{3}{2}x - 6\). Adding the term \(\frac{x}{2}\) in this part needs us to pay attention to fractions.

Remember, distributing involves multiplying each term inside the parentheses by the factor outside it. This ensures we preserve the equality of the equation while eliminating parentheses and preparing the equation for the next steps.

Identities and Contradictions

Understanding identities and contradictions is crucial when determining the nature of solutions for an algebraic equation. An **identity** occurs when an equation holds true for all combinations of variables within its domain.

• In the given equation, after simplifying, we found that \(2x - 6 = 2x - 6\) simplifies to \(-6 = -6\), an always-true statement that shows us any value of \(x\) will satisfy this equation.

If this simplified to something false—like \(0 = 1\)—it would be a **contradiction**, indicating there is no possible value of \(x\) to make the equation true. Recognizing when an equation is an identity or a contradiction assists in understanding whether solutions exist and how many there are.

Combining Like Terms

Combining like terms is an essential process in simplifying algebraic expressions and solving equations. It involves bringing together terms that have identical variable components, making it easier to see the underlying structure of the equation.

• For our exercise, after distribution, we had \(\frac{3}{2}x + \frac{x}{2}\) on the right side.
• To combine these, we used a common denominator and found \((\frac{3}{2} + \frac{1}{2})x = 2x\).

This step simplifies the equation to \(2x - 6 = 2x - 6\). It's important because it helps reduce complexity, allowing us to more easily identify patterns like in identities or contradictions, ultimately making the problem straightforward to solve.