Problem 17 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the equation is an identity, a conditional equation, or a contradiction. $$3+\frac{1}{x+1}=\frac{4 x}{x+1}$$

Step-by-Step Solution

Answer

The given equation is a conditional equation as it holds true for x = 4 only.

Step 1: Simplify the Equation

First, we must simplify the given equation to its most basic form. Let's multiply through the equation by \(x + 1\), doing so will also get rid of the fraction:\n\n\(3(x+1) + 1 = 4x\)\n\nThis simplifies to\n\n\(3x + 3 + 1 = 4x\)\n\nwhich further simplifies to\n\n\(3x + 4 = 4x\).

Step 2: Solve for x

Let's get x on one side of the equation to clarify things..\n\nBy subtracting \(3x\) from both sides, we obtain\n\n\(4 = x\).

Step 3: Classify the Equation

Now we have a value for x (x = 4) that makes the original equation true. So because our equation has one solution, it is a conditional equation - it holds true only for x = 4 and no other values of x.

Key Concepts

Identity

Identities are equations that are true for all possible values of the variable involved. Think of them as universally true statements. For example, the equation \( a + b = b + a \) is always true, regardless of the values of \(a\) and \(b\).

In the context of the given exercise, we are checking if the equation is an identity. If it were, no matter what value \( x \) takes, both sides of the equation \( 3 + \frac{1}{x+1} = \frac{4x}{x+1} \) would equate perfectly. However, during the solution process, we have discovered that this equation holds true only for \( x = 4 \). Thus, it is not an identity, because it does not apply universally.

Contradiction

A contradictory equation is one that is never true, regardless of what value is plugged in for the variable. It essentially has no solution. A classic example would be \( x + 1 = x + 2 \), which is clearly impossible since there are no numbers that satisfy this equation.

In our exercise, if after simplification and manipulation, both sides of the equation stood as something impossible like \( 0 = 1 \), we'd classify it as a contradiction. However, this equation gave a specific solution for \( x \), which means it is not contradictory. This single solution defines it not as a contradiction, but rather as a special type of equation known for having one solution.

Solving Equations

Solving equations involves finding the value(s) of the unknown variable that satisfy the equation. This is a central skill in algebra and is used across various mathematical problems.

In our exercise, we transformed the equation \( 3 + \frac{1}{x+1} = \frac{4x}{x+1} \) by multiplying both sides by \(x + 1\) to eliminate the fractions. This left us with \( 3x + 4 = 4x\). From here, we solved for \(x\) by rearranging terms to find \( x = 4 \). Each step in solving helps to zero in on the solution, serving as a crucial skill for understanding more complex mathematical concepts.

Simplification

The simplification process involves transforming an equation into its simplest form without changing its solution. This makes finding the solution easier.

For our equation, \(3 + \frac{1}{x+1} = \frac{4x}{x+1}\), simplification began by eliminating the fractions by multiplying through by \(x + 1\). This common factor effectively "cleared" the fraction, simplifying our task. We then combined like terms to reach \(3x + 4 = 4x\). Such transformations often reveal the nature of the solution more directly and clearly, making it easier to solve the equation accurately and efficiently.