Problem 40 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Perform the indicated operations and simplify. \(\frac{2+\frac{2}{x}}{x-\frac{2}{x}}\)

Step-by-Step Solution

Answer

The simplified expression is: \(\frac{2x+2}{x^2-2}\)

Step 1: Find a common denominator

In order to combine the fractions in the numerator and the denominator, we need to find a common denominator. In this case, the common denominator will be x, since it is the denominator of the two fractions inside the expression.

Step 2: Combine the fractions in the numerator

Now that we've found the common denominator (x), we will rewrite both terms in the numerator with this common denominator and combine them: \[\frac{2}{1}+\frac{2}{x}=\frac{2x}{x}+\frac{2}{x}\] So, the combined fraction in the numerator is: \[\frac{2x+2}{x}\]

Step 3: Combine the fractions in the denominator

We will do the same for the denominator: \[x-\frac{2}{x}=\frac{x^2}{x}-\frac{2}{x}\] So, the combined fraction in the denominator is: \[\frac{x^2-2}{x}\]

Step 4: Replace the original fractions in the expression

Now we will replace the original fractions in the expression with the combined fractions we found in Steps 2 and 3: \[\frac{\frac{2x+2}{x}}{\frac{x^2-2}{x}}\]

Step 5: Flip and multiply

To divide two fractions, we flip the second fraction (the one being divided) and multiply: \[\frac{2x+2}{x}\times\frac{x}{x^2-2}\]

Step 6: Simplify

Now we can simplify the expression by canceling the common factor x in the numerator and the denominator: \[\frac{2x+2}{x^2-2}\] The simplified expression is: \[\frac{2x+2}{x^2-2}\]

Key Concepts

Common Denominator

When we have fractions with different denominators, like \( \frac{2}{1} \) and \( \frac{2}{x} \), we need a common denominator to combine them. Choosing a common denominator allows us to rewrite each fraction with the same bottom number so they can be easily added or subtracted. Here, the common denominator is \( x \), because one of the fractions already has this as its denominator.
This means we rewrite both fractions:

• \( \frac{2}{1} = \frac{2x}{x} \) - by multiplying top and bottom by \( x \).
• \( \frac{2}{x} \) stays the same as it already has \( x \) as its denominator.

This step ensures all parts of the expression are aligned for combination.

Combine Fractions

Once the fractions have the same denominator, they can be easily combined into a single fraction. We just add or subtract the numerators while keeping the common denominator. In the given example, we have:- Numerator: \( \frac{2x}{x} + \frac{2}{x} = \frac{2x + 2}{x} \)- Denominator: \( \frac{x^2}{x} - \frac{2}{x} = \frac{x^2 - 2}{x} \)The operation of combining involves simple addition or subtraction of the numerators. Remember,

• Keep the denominator consistent.
• Just focus on the numerators for the operation.

This step transitions complex terms into single, more manageable fractions.

Flip and Multiply

Dividing by a fraction can feel tricky, but there's a straightforward method known as 'flip and multiply'. Here’s how it works:To divide \( \frac{\frac{2x+2}{x}}{\frac{x^2-2}{x}} \), we take the second fraction, flip it upside down, and multiply:- Flip: \( \frac{x^2-2}{x} \) becomes \( \frac{x}{x^2-2} \)- Multiply: \( \frac{2x+2}{x} \times \frac{x}{x^2-2} \)Flipping makes the division operation into a simpler multiplication problem, which is usually easier to handle. Make sure:

• The multiplication is performed across numerators and denominators.
• Any subsequent simplification occurs only after multiplication.

This strategy simplifies complex divisions into manageable calculations.

Algebraic Expression Simplification

After flipping and multiplying, the next step is simplifying the expression. This means reducing it to its simplest form. Simplifiers look for common factors that can be canceled between the numerator and the denominator. Here we simplify:\[ \frac{2x+2}{x^2-2} \]Look for common factors. Often terms in the numerator and denominator may have common elements that can be reduced.- Check the expression for any shared factors.- In our example, \( x \) cancels out on both sides since it shows up once in both places.This leaves us with a streamlined and simplified expression. Simplicity helps in reducing potential errors, making the expressions easier to work with in further calculations or evaluations.