Problem 44 - College Algebra

Question

Add or subtract as indicated. $$ \frac{4}{x}-\frac{3}{x+3} $$

Step-by-Step Solution

Answer

The simplified sum of the given fractions is \( \frac{x+12}{x^2+3x} \)

Step 1: Identify the Common Denominator

Since both fractions in the expression need to be either added or subtracted, it's necessary for their denominators to be the same, so that the expression can be simplified further. In this case the common denominator for these fractions will be their product because \(x\) and \(x+3\) have no common factors other than 1. So the common denominator is \(x*(x+3) = x^2 + 3x\)

Step 2: Rewrite the Fractions

To find an equivalent fraction with the common denominator, multiply both the numerator and the denominator of the first fraction by \(x+3\) and of the second fraction by \(x\). This gives: \[\frac{4(x+3)}{x^2 + 3x} - \frac{3x}{x^2+3x}\]

Step 3: Simplify the Expression

These fractions can now be combined since they have the same denominator. This is done by subtracting the numerators. That gives: \[\frac{4x+12-3x}{x^2+3x} = \frac{x+12}{x^2+3x}\]

Key Concepts

Common Denominator

When you add or subtract fractions, they must have the same denominator. This is referred to as a 'common denominator.' Without a common denominator, it's like trying to mix apples and oranges. In this exercise, the fractions given are \( \frac{4}{x} \) and \( \frac{3}{x+3} \).

Since the denominators in these fractions are different, you need to find a common ground. The easiest way to find a common denominator is to multiply the different denominators together.

• Multiply \( x \) and \( x+3 \) to get \( x(x+3) = x^2 + 3x \), which becomes the common denominator.

This process transforms the fractions so they can be combined or compared. Like having both friends at the same meeting spot, once you have a common denominator, it's much easier to proceed with addition or subtraction.

Equivalent Fractions

Creating equivalent fractions is the next step once you've identified a common denominator. An equivalent fraction is a fraction that represents the same value but has a different numerator and denominator.

Here’s how you do it for the exercise:

• For \( \frac{4}{x} \), multiply both the numerator and denominator by \( x+3 \) to adapt to the new denominator, resulting in \( \frac{4(x+3)}{x^2 + 3x} \).
• For \( \frac{3}{x+3} \), multiply both the numerator and denominator by \( x \) to match the denominator, resulting in \( \frac{3x}{x^2 + 3x} \).

These equivalent fractions now share the same denominator, \( x^2 + 3x \), allowing you to move to the next step, which involves combining the fractions with ease.

Simplifying Expressions

Simplifying an expression is like cleaning up your room; it makes everything clearer and more organized. Once you have found equivalent fractions with a common denominator, simplifying means bringing the terms together neatly.

For the exercise, you need to combine the numerators of the equations since the denominators are already the same. This process is straightforward:

• Subtract the numerators: \( 4(x+3) - 3x \) becomes \( 4x + 12 - 3x \).
• Combine like terms: \( 4x - 3x + 12 = x + 12 \).

Now, put it all together over the common denominator: \( \frac{x+12}{x^2 + 3x} \).
This is your simplified expression. The beauty here is that you've transformed two separate fractions into one cohesive expression, allowing for an easier and more precise representation of the mathematical sentence.