Problem 11 - Algebra 2

Question

Simplify each sum. \(\frac{d-3}{2 d+1}+\frac{d-1}{2 d+1}\)

Step-by-Step Solution

Answer

The simplified form of the sum \(\frac{d-3}{2 d+1}+\frac{d-1}{2 d+1}\) is \(\frac{2d-4}{2 d+1}\).

Step 1: Identify common denominator

The given fractions are \(\frac{d-3}{2 d+1}\) and \(\frac{d-1}{2 d+1}\). Both fractions have the same denominator \((2 d+1)\). This makes our work easier because we won't need to find a least common denominator.

Step 2: Add the numerators

Since both fractions have the same denominator, we can add the fractions by simply adding the numerators and putting the sum over the common denominator. So, \(\frac{d-3}{2 d+1}+\frac{d-1}{2 d+1}=\frac{(d-3)+(d-1)}{2 d+1}\). This simplifies to \(\frac{2d-4}{2 d+1}\).

Step 3: Check and simplify fraction

We now have the fraction \(\frac{2d-4}{2 d+1}\). To simplify this, check if the numerator and denominator have any common factors that we can cancel out. In this case, they don't. So, the simplified form of the given sum is \(\frac{2d-4}{2 d+1}\).

Key Concepts

Common Denominator

When dealing with fractions, the denominator is the number or expression on the bottom. For fraction addition, like in the exercise \(\frac{d-3}{2d+1}+\frac{d-1}{2d+1}\), it is essential to have a common denominator. In this example, we are in luck because both fractions already have the same denominator, \(2d+1\). This simplifies our work.

• A common denominator allows you to directly add the fractions.
• No need to find a least common denominator means less calculation.

Understanding how a common denominator works is fundamental to manipulating and adding fractions effectively.

Fraction Addition

Adding fractions requires more than just summing the numerators. When fractions share a common denominator, as in the example, it's straightforward. You only need to add the numerators. In our exercise, \[ \frac{d-3}{2d+1} + \frac{d-1}{2d+1} = \frac{(d-3)+(d-1)}{2d+1} \]

• The denominator remains \((2d+1)\) because both fractions have the same denominator.
• The numerator expression simplifies as you combine like terms.

Fraction addition becomes simple when you add the numerators and maintain the denominator if they are already common. This method balances the addition operation by focusing on similar terms.

Simplifying Fractions

Simplification mainly involves reducing a fraction to its simplest form. After addition, the result is \(\frac{2d-4}{2d+1}\). The next step is to determine if you can reduce this fraction by canceling out any common factors

• Look at the numerator \(2d-4\) and the denominator \(2d+1\). Do they share any factors?
• If there's a common factor in the numerator and denominator, you can simplify further by dividing both by that factor.

In our exercise, \(2d-4\) and \(2d+1\) do not have common factors to cancel. Thus, \(\frac{2d-4}{2d+1}\) is already in its simplest form. Simplifying fractions helps get a more readable and easy-to-understand result.

Like Denominators

"Like denominators" simply means that two or more fractions share the same denominator. This term is very helpful when performing addition or subtraction of fractions, as it allows us to directly work on the numerators without changing the denominators first. In the problem:\[ \frac{d-3}{2d+1} + \frac{d-1}{2d+1} \]Both fractions already have like denominators \((2d+1)\).

• You can combine them directly by adding the numerators \((d-3)+(d-1)\).
• Like denominators simplify many operations, such as finding equivalent fractions and making comparisons.

Understanding "like denominators" is a handy technique in making the computation of fractions smoother and more manageable.