Problem 52 - 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. $$3(x+2)-5(x+2)=-2 x-4$$

Step-by-Step Solution

Answer

The equation is an identity with the solution set as all real numbers, \(\{ x | x \in \mathbb{R} \}\).

Step 1: Expand Both Sides

First, use the distributive property to expand both sides of the equation. The equation is initially given as:\[3(x+2)-5(x+2)=-2x-4\]Distribute the numbers 3 and -5 over the parentheses:\[3x + 6 - 5x - 10 = -2x - 4\]

Step 2: Combine Like Terms

Next, combine like terms on each side of the equation. On the left side, combine the \(3x\) and \(-5x\), and \(6\) and \(-10\):\[-2x - 4 = -2x - 4\]

Step 3: Simplify the Equation

Simplify the equation to see if both sides are equal. We notice that after combining like terms, both sides of the equation are identical:\[-2x - 4 = -2x - 4\]This means the equation holds true for all values of \(x\).

Step 4: Identify the Type of Equation

Since the simplified form \(-2x - 4 = -2x - 4\) is true for any value of \(x\), it is an identity. An identity is an equation that is true for all values of the variable.

Step 5: Solution Set

The solution set for an identity is all real numbers, \(\mathbb{R}\). Because the equation is always true, there are no restrictions on the values of \(x\). You can represent this as:Solution Set: \(\{ x | x \in \mathbb{R} \}\).

Key Concepts

Contradiction

In mathematics, a contradiction arises when you have an equation where no possible value for the variable can make it true. This means the equation is internally inconsistent. It’s like saying two plus two equals five—it simply never happens.When working with equations, if you simplify both sides and end up with terms that logically cannot work together, such as a statement like \(0 = 3\), you have found a contradiction.Here’s the nifty thing: Contradictory equations have no solutions! That means the solution set is empty because no value will ever satisfy the equation. Mathematically, we represent this empty set as:

• Empty Set: \(\emptyset\) or \(\{ \}\)

It's good to remember that contradictions hint at an inconsistency in the initial setup or assumptions of the equation.

Solution Set

Every equation's job is to find which values of a variable, typically \(x\), will satisfy the equation. These satisfying values of \(x\) are what make up the "solution set."Each type of equation gives us a different kind of solution set:

• Contradiction: No solutions, so the solution set is empty.
• Identity: All real numbers are solutions, hence the solution set is all real numbers \(\mathbb{R}\).
• Conditional Equation: A specific set of values that satisfy the equation.

For an identity equation like the one in the exercise, the solution set is particularly interesting because it includes every possible number. In simpler terms, it doesn't matter what number you substitute for \(x\), the equation holds true. We can represent this abundant solution set as:

• Solution Set for Identity: \(\{ x | x \in \mathbb{R} \}\)

Remember, understanding the kind of equation you're dealing with can quickly inform you of the nature of its solution set.

Distributive Property

The distributive property is a key concept in algebra that simplifies expressions and equations. It helps you multiply a single term by each term inside a set of parentheses. It's written mathematically as \( a(b+c) = ab + ac \).Let's break down how this worked in the given equation. You have:

• \(3(x+2)\) becomes \(3 \cdot x + 3 \cdot 2\), which simplifies to \(3x + 6\).
• \(-5(x+2)\) turns into \(-5 \cdot x - 5 \cdot 2\), simplifying it to \(-5x - 10\).

Once you apply the distributive property, the next step is to combine like terms, which reduces the equation to a simpler form. This process is especially useful for solving equations or simplifying expressions to make them more manageable.The distributive property doesn't just help in simplifying equations, but it's a foundational tool that supports many algebraic processes, ensuring clarity and consistency in calculations.