Problem 22 - Algebra 1

Question

Simplify the expression. $$\frac{x}{x+2} \div \frac{x+5}{x+2}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{x}{x+5}\) where \(x \neq -5\).

Step 1: Change Division To Multiplication

The division of two fractions can be transformed into the multiplication by inverting the second fraction. So, the expression \(\frac{x}{x+2} \div \frac{x+5}{x+2}\) would become \(\frac{x}{x+2} * \frac{x+2}{x+5}\). In this step, the reciprocal of the divisor \(\frac{x+5}{x+2}\) is calculated, which is \(\frac{x+2}{x+5}\). After that, the initial division operation is changed to multiplication.

Step 2: Simplify

Next, cancel the common factors in the numerator and the denominator. Here the term \((x+2)\) is common to both numerator and denominator. After canceling out, the expression becomes \(\frac{x}{x+5}\).

Step 3: Check

Always remember to check the domain of the expression. Here the denominator shouldn't be zero, therefore \(x \neq -5\).

Key Concepts

Division of Fractions

Understanding how to divide fractions is crucial for solving various algebraic expressions and equations. When you're dividing by a fraction, you essentially multiply by its reciprocal. Wondering what a reciprocal is? It's simply flipping the numerator and the denominator of a fraction around.

Here's how you can convert division into multiplication:

• Identify the divisor, which is the second fraction in the division.
• Take the reciprocal by swapping the positions of the numerator and denominator.
• Change the division sign to multiplication.

After these steps, what was once a division problem becomes a straightforward multiplication.In our exercise, we started with \( \frac{x}{x+2} \div \frac{x+5}{x+2} \).Upon converting the division into multiplication, it became:\( \frac{x}{x+2} \times \frac{x+2}{x+5} \).This swap simplifies the operations with fractions, making it much easier to handle them in expressions.

Multiplication of Fractions

Multiplying fractions is quite simple once you have converted the division into multiplication. If you are multiplying two fractions together:

• Multiply the numerators together to get a new numerator.
• Multiply the denominators together to get a new denominator.

In the equation \( \frac{x}{x+2} \times \frac{x+2}{x+5} \), we multiply directly:

• The numerators: \( x \times (x+2) \)
• The denominators: \( (x+2) \times (x+5) \)

However, before you multiply everything out, it's always wise to search for common factors. This is because simplifying beforehand often saves time and makes your results easier to manage. In this example, we don’t directly multiply everything out because there’s a common factor we can cancel.

Canceling Common Factors

Canceling common factors is an essential skill for simplifying fractions. It helps to make expressions cleaner and reasonably straight-forward. You might wonder why canceling is even allowed. Well, it’s because of the principle that any number divided by itself equals one. When simplifying the expression \( \frac{x}{x+2} \times \frac{x+2}{x+5} \), you can spot that \( (x+2) \) appears as both a numerator and a denominator. These terms can "cancel" each other out:

• Identify the (x+2) as a common factor.
• Cancel (x+2) in both the numerator and the denominator.
• Your expression now simplifies to \( \frac{x}{x+5} \).

Be cautious when canceling factors and ensure they're common and non-zero, as division by zero is undefined. This process of canceling common factors not only simplifies the fraction, but also avoids unnecessary complications in handling larger numbers.