Problem 60 - 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. $$0.5(x-2)+12=0.5 x+11$$

Step-by-Step Solution

Answer

The equation is an identity; the solution set is all real numbers.

Step 1: Simplify the equation

We have the equation \(0.5(x-2) + 12 = 0.5x + 11\). Start by expanding the left side: \(0.5(x-2) = 0.5x - 1\). So the equation becomes \(0.5x - 1 + 12 = 0.5x + 11\). Simplify the left side to get \(0.5x + 11 = 0.5x + 11\).

Step 2: Check for equality

Now compare both sides of the simplified equation: \(0.5x + 11 = 0.5x + 11\). Since both sides are identical, the equation is an identity.

Step 3: Determine the solution set

An identity means that the equation is true for any value of \(x\). Thus, the solution set is all real numbers.

Step 4: Graphical representation

Graphically, both sides correspond to the same line. For any value of \(x\), plotting each side will result in overlapping lines, confirming that they are equal for all \(x\).

Key Concepts

Identity Equation

An identity equation is a fascinating type of equation where both sides are the same, making it true for every possible value of the variable involved. In simpler terms, it is like stating a fact that applies universally. For example, in the equation \( 0.5(x-2) + 12 = 0.5x + 11 \), simplifying it leads to \( 0.5x + 11 = 0.5x + 11 \). Both sides are identical, indicating an identity equation.

Here’s what makes identity equations special:

• They illustrate mathematical facts or truths.
• There is no need to find specific values for the variable since it holds true for all possible values.
• They are often used to simplify expressions or solve more complex mathematical problems.

Whenever faced with an identity equation, remember that it's true for all values, showcasing the universality of certain mathematical truths.

Solution Set

The solution set of an equation is all the values that satisfy the equation. For identity equations, like our example, the solution set is especially broad: it includes all real numbers. This is because the equation holds true no matter what value we substitute for the variable.

Understanding the solution set involves:

• Recognizing that identity equations, such as \( x = x \), are valid for every real number \( x \).
• Knowing that there is no restriction or boundary on the values \( x \) can take.
• Accepting that the answer "all real numbers" means the equation is universally applicable.

While some equations only hold for specific numbers (conditional equations) or not at all (contradictions), identity equations have the most inclusive and wide-ranging solution sets.

Graphical Representation

Graphical representation provides a visual understanding of an equation. For an identity equation, such as \( 0.5(x-2) + 12 = 0.5x + 11\), graphing both sides of the equation demonstrates that they overlap completely on the coordinate plane.

Here's how to visualize graphically:

• Plot both sides of the equation as separate lines or curves using the same coordinate axes.
• In the case of an identity equation, you'll find that one line overlaps the other entirely.
• This overlap confirms that every point on one line is also on the other, reinforcing that the equation holds for all real number \( x\).

Graphical representation is helpful because it provides clear, visual proof of the equation's properties, confirming whether two expressions are equal across all possible values.