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

Step-by-Step Solution

Answer

Identity; solution set is all real numbers (\(\mathbb{R}\)).

Step 1: Simplify the Equation

First, simplify both sides of the equation \(0.5(x-2) + 12 = 0.5x + 11\). Distribute the \(0.5\) on the left-hand side: \(0.5x - 1 + 12 = 0.5x + 11\). This simplifies further to \(0.5x + 11 = 0.5x + 11\).

Step 2: Analyze Variables Canceling Out

Subtract \(0.5x\) from both sides to determine whether the equation is an identity, contradiction, or conditional: \(11 = 11\). This shows that after canceling the variable terms, both sides are equal.

Step 3: Classify the Equation

Since the simplified equation \(11 = 11\) is always true and holds for any value of \(x\), the original equation is classified as an identity.

Step 4: Determine the Solution Set

Because an identity is true for all values of \(x\), the solution set for \(0.5(x-2)+12=0.5x+11\) is all real numbers \((\mathbb{R})\).

Step 5: Verification with Graph or Table

Graph the lines \(y = 0.5(x-2) + 12\) and \(y = 0.5x + 11\). Observe that the lines overlap completely on the graph, confirming the solution set is all real numbers.

Key Concepts

Identity

An identity equation is a type of equation that is true for all possible values of the variable. It means that no matter what value you substitute into the equation, it will always hold as true. In the context of our original exercise, after simplifying the equation

• Distributing the 0.5: \(0.5x - 1 + 12 = 0.5x + 11\)
• Further simplification leads to: \(0.5x + 11 = 0.5x + 11\).

When both sides are identical, removing the variable term leaves a true statement: \(11 = 11\).
This confirms that the equation is an identity.
Identity equations result in statements like \(0 = 0\) or any number equaling itself, which are valid universally.
They are crucial because they show that the equality holds no matter what value is chosen for the variable.
Identities are foundational, helping to establish truths in mathematics applicable across all scenarios.

Contradiction

Contradiction equations are intriguing because they always lead to a false statement, regardless of the variable value you plug in. Contrary to an identity, a contradiction happens when, upon simplification, the equation loses the variable but results in a statement that is inherently false. If, when solving, you end up with something entirely conflicting like \(0 = 1\), this is a clear sign of a contradiction.
Such equations will never have a valid solution set because no value for the variable can satisfy them. Had our original equation reduced to a contradiction, it would mean there was a fundamental error in expecting a balance in the, equation in the first place.
Contradiction equations alert us when assumptions or parts of equations are incorrect or incompatible.

Conditional Equation

Conditional equations hold their truth only under specific conditions or for certain values of the variable. In simple terms, these are equations that have a unique solution or a set of solutions, beyond which the equation turns false. This type of equation differs from identity equations, which are always true no matter what.
Imagine an equation that simplifies to give you one particular solution. For example, suppose after simplifying the equation, you found it boils down to \(x = 5\). This means only when \(x\) is 5 does the equation hold true. At any other value, it becomes false.

• Conditional equations highlight specific circumstances or ranges where a mathematical balance is optimized.
• They are prevalent in various applications, where precise conditions must be met, such as physics or engineering problems.

It's compelling to know what categorizes conditional equations as they potentially unlock specific results rather than a general truth.

Solution Set

The solution set of an equation encompasses all possible values of the variable that make the equation true. Identifying the solution set is a critical task in solving equations, as it essentially tells us

• the range of numbers that satisfy the equation efficiently.
• In our problem, where the equation was classified as an identity, the solution set is all real numbers, denoted as \(\mathbb{R}\).

When solving an equation, you're frequently determining a solution set. It may include:

• A single value (like \(x = 3\) for unique solutions).
• No value at all (as seen with contradictions).
• The entirety of possible values (in the case of identities).

Understanding what constitutes a solution set helps visualize the range or limitations of an equation, enhancing comprehension for more complex problems.
The graphical or tabular representation further aids by providing a visual that confirms the solutions obtained.