Problem 82 - Algebra A Combined Function

Question

For what value of \(x\) is \(\frac{x}{x-1}\) in proportion to \(\frac{x+1}{x} ?\) Explain your result.

Step-by-Step Solution

Answer

No real value of \( x \) satisfies the given proportion.

Step 1: Understand the concept of proportion

Two fractions are in proportion if their cross products are equal. If \( \frac{a}{b} = \frac{c}{d} \), then \( a \times d = b \times c \). This is the method we will use to solve the given problem.

Step 2: Set up the proportion equation

We are given that \( \frac{x}{x-1} \) is in proportion to \( \frac{x+1}{x} \). This implies that \( \frac{x}{x-1} = \frac{x+1}{x} \), which can be expressed as \( x \times x = (x-1) \times (x+1) \) after setting up the cross products.

Step 3: Simplify the equation

Expand the cross product equation \( x \times x = (x-1)(x+1) \). This results in \( x^2 = x^2 + x - x - 1 \). Here, the terms \( +x \) and \( -x \) cancel each other, resulting in \( x^2 = x^2 - 1 \).

Step 4: Solve for x

Since both sides of the equation \( x^2 = x^2 - 1 \) have a common term of \( x^2 \), subtract \( x^2 \) from both sides, which gives \( 0 = -1 \). This result is a contradiction, meaning no real number \( x \) will satisfy the given proportion.

Key Concepts

Proportion

The concept of proportion is a fundamental idea in algebra and mathematics in general. Proportions are used to show that two ratios or fractions are equivalent. In algebra, if you have two fractions, \( \frac{a}{b} \) and \( \frac{c}{d} \), they are said to be in proportion if they represent the same value; this is expressed as \( \frac{a}{b} = \frac{c}{d} \).

• Proportions are a key tool in solving real-world problems where relationships between quantities are important.
• When dealing with proportions, it helps to cross multiply to check if the fractions are equivalent.

In the problem, we are asked to determine if \( \frac{x}{x-1} \) is in proportion to \( \frac{x+1}{x} \). If these two fractions form a proportion, then their cross products must be equal.

Cross Multiplication

Cross multiplication is a method used to solve equations involving proportions. This technique is valuable because it eliminates the fractions, allowing for simpler equation solving steps. In cross multiplication, you multiply the numerator of one fraction by the denominator of the other fraction and vice versa.
For example, given \( \frac{a}{b} = \frac{c}{d} \), you multiply across the equal sign to get \( a \times d = b \times c \). This is called the cross product.

• This method safeguards against inaccuracies by eliminating the fractions early in the process.
• Cross multiplying aids in determining whether two fractions are equivalent by comparing their products.

In the exercise, the equality \( x \times x = (x-1) \times (x+1) \) was derived from cross multiplying, setting the stage for further simplification.

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. They play a vital role in algebra because they extend the concept of fractions to more complex expressions. Rational expressions can be simplified, added, subtracted, multiplied, and divided, much like standard fractions.

• It's crucial to remember that the denominator of a rational expression cannot be zero, as division by zero is undefined.
• When solving problems involving rational expressions, identifying and cancelling common factors simplifies the work.

In the context of our exercise, both \( \frac{x}{x-1} \) and \( \frac{x+1}{x} \) are rational expressions. The task of solving proportions with these expressions involves ensuring they are defined by making sure the denominators are not zero.

Equation Solving

After using cross multiplication, the next step is solving the resulting equation. In algebra, solving an equation involves finding the value of the variable that makes the equation true. Simplifying the equation and performing operations like addition, subtraction, multiplication, or division are typical steps.

• It's crucial to perform the same operation on both sides of the equation to maintain equality.
• Sometimes, equations reveal inconsistencies, such as a contradiction or an identity.

In the problem, simplifying \( x^2 = x^2 - 1 \) resulted in a contradiction, \( 0 = -1 \). This indicates that no real number can satisfy the equation, highlighting that not all algebraic problems have real number solutions.