Problem 67 - College Algebra Essentials

Question

Solve equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. \(\frac{2 x}{x-3}=\frac{6}{x-3}+4\)

Step-by-Step Solution

Answer

The solutions is \(x = 3\) and the given equation is an inconsistent equation.

Step 1: Simplify the Equation

First, both terms on the left and right-hand side of the equation have a common denominator of \(x - 3\). This allows us to simplify the equation by multiplying the entire equation through by \(x - 3\) to eliminate the fraction: \(2x = 6 + 4 \cdot (x - 3)\).

Step 2: Expand and Simplify

Next, expand the brackets on the right-hand side of the equation and then simplify: \(2x = 6 + 4x - 12\), which simplifies to \(2x = 4x - 6\).

Step 3: Solve for x

Now, solve the equation for \(x\) by re-arranging terms: \(-2x= -6\), which then simplified to \(x = 3\).

Step 4: Determine the Type of Equation

Test the solution by injecting \(x = 3\) back into the original equation: \(\frac{2 \cdot 3 }{3 -3} = \frac{6}{3 - 3} + 4\). As we can see that both sides of the equation are undefined (dividing by zero), hence the solution does not satisfy our original equation. Thus, the equation is classified as an inconsistent equation.

Key Concepts

Conditional Equation

When you're solving algebraic equations, one of the types you might encounter is a conditional equation. A conditional equation is true only for certain values of the variable. These are not universal truths but rather situations that depend on specific conditions being met.

For instance, the equation \(x - 3 = 0\) is a conditional equation, because it is only true when \(x\) equals 3. When you're solving a problem, and you arrive at a valid solution that satisfies the original equation for some specific value(s), you've found the conditions that make the equation true.

Inconsistent Equation

Sometimes, you may come across an equation that has no solution at all. This type of equation is known as an inconsistent equation. No matter what value you assign to the variable, the equation will never be true.

An example of this situation is seen in the original problem provided. When you find that substituting the solution back into the equation results in an impossible scenario (like dividing by zero), it indicates that the equation does not hold true for any value and is inconsistent. Understanding this concept can save you time—you'll know that, if the math leads you to an inconsistency, you won't need to look any further for a solution.

Identity Equation

In contrast to conditional and inconsistent equations, an identity equation is true for all values of the variable. This means it's an equation that essentially states a fact about numbers, such as \(2(x + 3) = 2x + 6\).

No matter what value you substitute for \(x\), both sides of the equation will remain equal. This is because the equation is balanced and simplified on both sides. When you come across this kind of equation in your work, know that you're dealing with an algebraic truth rather than a condition or an inconsistency.

Algebraic Fractions Simplification

Working with algebraic fractions can get tricky, but simplifying them often makes the task easier. Algebraic fractions simplification involves reducing the complexities of fractions by either finding a common denominator or canceling out common factors.

To simplify, you may multiply each term by the least common denominator to eliminate the fractions as seen in the provided exercise, or reduce the expression to its simplest form by dividing both numerator and denominator by their greatest common factor. By doing so, you clarify the equation, which allows you to see the path to the solution more clearly.