Problem 136 - College Algebra

Question

This will help you prepare for the material covered in the next section. $$\text { Simplify: } \frac{x+1}{x+3}-2$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{-x-5}{x+3}\)

Step 1: Identify the denominators involved

Here we have a fraction \(\frac{x+1}{x+3}\) and a whole number 2 (which can be thought of as \(\frac{2}{1}\)). So the two denominators are \(x+3\) and 1.

Step 2: Find the Common Denominator

The common denominator of \(x+3\) and 1 would simply be \(x+3\), as any number is divisible by 1. The equivalent of 2 with this common denominator would be \(\frac{2(x+3)}{x+3}\) or \(2x+6\). Hence we rewrite the expression as \(\frac{x+1}{x+3}-\frac{2(x+3)}{x+3}\).

Step 3: Perform the Subtraction

Now that we have a common denominator, we can perform the subtraction. We subtract the numerator of the second fraction from the numerator of the first fraction while keeping the denominator the same. This results in \(\frac{x+1-2*(x+3)}{x+3}\).

Step 4: Simplify the Expression

By expanding the expression above, we obtain \(\frac{x+1-2x-6}{x+3} = \frac{-x-5}{x+3}\). This is the simplified form of the given expression.

Key Concepts

Fraction Subtraction

When dealing with fractions, subtracting them means focusing first on the numerators and the denominators. The key is to keep the denominators the same while subtracting the values on top (numerators). This is important because fractions represent parts of a whole, and altering the denominator during direct subtraction could lead to errors. In this exercise, the fraction subtraction occurs after rewriting both terms with a common denominator. The original expression is \( \frac{x+1}{x+3} - 2 \). To perform the subtraction, the integer 2 is rewritten as a fraction with the same common denominator, which transforms the problem into subtracting two fractions. This way, the operation becomes manageable and accurate.

Common Denominator

A common denominator is essential in both fraction addition and subtraction. It provides a common platform for these operations by making it possible to compare and combine fractions. To find a common denominator, you look for a number that each of your denominators can divide into without leaving a remainder. In this exercise, we have two distinct denominators: \(x+3\) and \(1\). Since any number can divide 1, the least common denominator here is \(x+3\). With this, the exercise turns a straightforward subtraction problem because both terms involved are changed to having \(x+3\) as their denominator.

• The expression \(2\) is transformed into \(\frac{2(x+3)}{x+3} \) to align with the fraction \(\frac{x+1}{x+3} \).
• The result is a new expression, \(\frac{x+1}{x+3} - \frac{2(x+3)}{x+3} \), now prepared for subtraction.

Understanding this concept unlocks the ability to solve many problems involving rational expressions.

Simplifying Expressions

Simplification is a critical step in algebra that involves reducing expressions to their simplest form. It makes equations cleaner, more understandable, and often easier to work with. After obtaining a common denominator and performing the subtraction, you simplify the expression to find the most reduced form possible. For the expression \(\frac{x+1 - 2(x+3)}{x+3}\), simplification involves distributing and combining like terms.

Here's how it's broken down:

• Distribute the multiplication in \(-2(x+3)\) to get \( -2x - 6 \).
• Combine all like terms in the numerator: \(x + 1 - 2x - 6 = -x - 5\).
• The fraction then looks like \(\frac{-x - 5}{x+3}\).

This final step is crucial as it provides a nicely reduced expression, revealing the fundamental relationship between variables. Remember, breaking down complex expressions makes them more understandable and helps in uncovering hidden patterns.