Problem 71 - College Algebra Essentials

Question

Combine the types of equations we have discussed in this section. Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. \(\frac{2}{x-2}=3+\frac{x}{x-2}\)

Step-by-Step Solution

Answer

The solution \(x = 2\) is not possible since in the beginning, it was determined that x cannot be 2. This means the original equation has no solution; it is an inconsistent equation.

Step 1: Identify the Undefined Values

As this equation has denominators that involve 'x', start by identifying the values that make the denominator equal to zero. Solving \(x - 2 = 0\) , it gives \(x = 2\). Therefore, in this equation, x cannot be 2.

Step 2: Combine Similar Terms

Now, let's simplify the equation. Note that both fractions have the same denominator, therefore sums and differences of fractions can be combined using a common denominator. This simplifies the equation to \(\frac{2 - x}{x-2} = 3\).

Step 3: Multiply by the Denominator

To make the equation easier to solve, eliminate the denominator by multiplying both sides of the equation by \(x - 2\). It yields: \(2 - x = 3(x-2)\).

Step 4: Distribute and Simplify

Next, distribute the 3 to \(x - 2\), and simplify the equation. This results in \(2 - x = 3x - 6\). This equation can be further simplified to get \(4x = 8\).

Step 5: Solve for the Variable

Now, isolate 'x' by dividing both sides of the equation by 4. That gives \(x = 2\).

Key Concepts

Identity Equations

Identity equations are special kinds of equations that are true for all possible values of the variable involved. This implies that both sides of the equation are always equal, regardless of the value assigned to the variable. An example of an identity equation is one that simplifies to something like: \( x + 0 = x \). This statement is always true, no matter what number you substitute for \( x \).

Identity equations can often be identified during the process of solving. If you end up with a trivial truth like \( 0 = 0 \) after simplifying both sides, you've encountered an identity equation.

• True for all values (in its domain)
• Simplifies to an obvious truth
• No specific solution, because every value is a solution

Understanding identity equations is crucial because it helps pinpoint equations that always hold true—these are not dependent upon specific conditions or variables.

Conditional Equations

Conditional equations are equations that hold true only under certain conditions or for specific values of the variable. These differ from identity equations, as they do not hold true for every possible value. The true nature of these equations is found by solving them, just as in the exercise.In the exercise, solving \( \frac{2-x}{x-2} = 3 \) led us to a solution \( x = 2 \), which we further found to be an undefined value for the original equation and hence an extraneous solution.

• True only for specific values
• May have zero, one, or multiple solutions
• Simplified equations may sometimes yield extraneous solutions

Recognizing conditional equations involves checking if proposed solutions work within the original framework, ensuring they don't violate any conditions such as division by zero or undefined elements.

Inconsistent Equations

Inconsistent equations are equations that have no solution. When you work through an inconsistent equation, you often end up at a logical impossibility such as a contradiction. Consider an equation that simplifies down to something like \( 2 = 3 \). This is clearly false, and demonstrates why the equation is inconsistent.During the solving process, inconsistent equations suggest that no value of the variable can satisfy the condition set by the equation.

• Contradictory result upon simplification
• Leads to no solution
• Indicates a fundamental issue with the equation setup

Being able to identify inconsistent equations helps avoid wasting time attempting to find solutions where there are none, and instead focus on recognizing where errors or misunderstandings might have occurred.