Problem 39 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations and simplify. \(\frac{1+\frac{1}{x}}{1-\frac{1}{x}}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{x+1}{x-1}\).

Step 1: Identify sub-fractions

First, let's identify the sub-fractions in the given expression. There is one fraction in the numerator and another one in the denominator. \(\frac{1+\frac{1}{x}}{1-\frac{1}{x}}\) The sub-fractions are: Numerator sub-fraction: \(\frac{1}{x}\) Denominator sub-fraction: \(-\frac{1}{x}\)

Step 2: Simplify the numerator and denominator

Now, let's simplify the numerator and denominator by combining the sub-fractions with their respective constants. Numerator: \(1+\frac{1}{x}=\frac{1\times x}{x}+\frac{1}{x}=\frac{x+1}{x}\) Denominator: \(1-\frac{1}{x}=\frac{1\times x}{x}-\frac{1}{x}=\frac{x-1}{x}\) After simplifying, the expression becomes: \(\frac{\frac{x+1}{x}}{\frac{x-1}{x}}\)

Step 3: Divide the fraction expressions

Now that we have simplified both the numerator and the denominator, we can divide the expressions by multiplying the numerator by the reciprocal of the denominator. \(\frac{\frac{x+1}{x}}{\frac{x-1}{x}} = \frac{x+1}{x}\times\frac{x}{x-1}\)

Step 4: Simplify the expression

Next, let's simplify the expression by canceling out any common factors. In this case, we can cancel out 'x' from both the numerator and the denominator. \(\frac{x+1}{x}\times\frac{x}{x-1}=\frac{(x+1)\cancel{x}}{\cancel{x}(x-1)} = \frac{x+1}{x-1}\)

Step 5: Write the final simplified fraction

The final simplified expression is: \(\frac{x+1}{x-1}\)

Key Concepts

Algebraic Expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operators (like plus or minus). They are like sentences in math language. In our exercise, the expression

• \( \frac{1+\frac{1}{x}}{1-\frac{1}{x}} \)

is an example of a complex algebraic expression. Here, it includes a fraction as well as a variable \( x \).
Breaking down and analyzing each part of the expression helps us understand and simplify it. By simplifying, we make the expression easier to manage and work with.

Mathematical Operations

Mathematical operations, such as addition, subtraction, multiplication, and division, are fundamental in working with mathematical expressions. In this exercise, multiple steps illustrate these operations.
The initial task is to identify and simplify each sub-fraction in both the numerator and denominator.

• In the numerator, \( 1+\frac{1}{x} \), we combine the constant with the fraction: \( \frac{1\times x}{x}+\frac{1}{x}=\frac{x+1}{x} \).
• In the denominator, \( 1-\frac{1}{x} \), the same process applies, resulting in: \( \frac{1\times x}{x}-\frac{1}{x}=\frac{x-1}{x} \).

By practicing these operations, you build a strong foundation for manipulating algebraic expressions.

Rational Expressions

A rational expression is simply a fraction where the numerator and/or the denominator are algebraic expressions. Here, our task involved simplifying the rational expression \( \frac{\frac{x+1}{x}}{\frac{x-1}{x}} \).
One key operation is to multiply by the reciprocal. This turns the division into multiplication:\[ \frac{x+1}{x} \times \frac{x}{x-1} \]Next, we simplify by canceling common factors. In this case, the variable \( x \) is present in both the numerator and the denominator, allowing us to cancel it out:\[ \frac{(x+1)\cancel{x}}{\cancel{x}(x-1)} = \frac{x+1}{x-1} \]After simplifying, you're left with a simpler, equivalent fraction. Mastering these techniques is crucial for working effectively with fractions that have variables.