Problem 87 - College Algebra with Modeling and Visualization

Question

Simplify. $$ \frac{b}{2 b-4}-\frac{b-1}{b-2} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-\frac{1}{2}\).

Step 1: Find a Common Denominator

The denominators we have are \(2b - 4\) and \(b - 2\). Notice that \(2b - 4\) can be factored as \(2(b - 2)\). Thus, the common denominator is \(2(b - 2)\).

Step 2: Rewrite Each Fraction with a Common Denominator

For \(\frac{b}{2b - 4}\), it is already in terms of the common denominator. More explicitly, it is \(\frac{b}{2(b-2)}\).For \(\frac{b - 1}{b - 2}\), multiply both the numerator and denominator by 2 to get \(\frac{2(b-1)}{2(b-2)}\).

Step 3: Subtract the Fractions

Now both fractions are \(\frac{b}{2(b-2)}\) and \(\frac{2(b-1)}{2(b-2)}\). Subtract them to get:\[\frac{b - 2(b-1)}{2(b-2)}\]Simplify the numerator as follows:\[b - 2(b-1) = b - 2b + 2 = -b + 2\]So, the fraction becomes:\[\frac{-b + 2}{2(b-2)}\]

Step 4: Simplify the Result

There's nothing more to cancel out or simplify. Therefore, the simplified form of the expression is:\[\frac{2-b}{2(b-2)}\]To make it more consistent with factoring conventions, you might alternatively write it as:\[-\frac{b-2}{2(b-2)} = -\frac{1}{2}\]after cancelling out \(b-2\) from the numerator and denominator.

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in algebra. It involves reducing a fraction to its simplest form, which makes calculations easier and results clearer.

To simplify a fraction, the numerator and the denominator must be factored down to their smallest values such that they do not share any common factors other than 1. If they do share common factors, these can be divided out of both the numerator and denominator.

For example, in the original exercise, our goal is to simplify the expression:

• Identify any common factors between numerator and denominator.
• Cancel out common factors to reduce the fraction.
• Repeatedly factor if needed until the simplest form is reached.

It is crucial because a simplified fraction is easier to interpret and use in further calculations.

Common Denominators

Finding a common denominator is a critical part of adding or subtracting fractions. The common denominator allows you to combine fractions by ensuring they have the same base.

In the step-by-step solution given, the key to solving the problem lies in identifying a common denominator for the fractions \ \(\frac{b}{2b - 4}\) and \ \(\frac{b-1}{b-2}\). This process involves:

• Factoring existing denominators when possible, as demonstrated with \(2b - 4\) becoming \(2(b - 2)\).
• Determining a common denominator by choosing one that can accommodate all terms involved, which is \(2(b - 2)\) in this case.

This step is essential for the next stage where fractions are rewritten with the common denominator. This makes it feasible to perform operations like addition and subtraction since the denominators match.

Factoring

Factoring is the process of breaking down expressions into their multiplicative components. It is a cornerstone technique in algebra and helps simplify expressions significantly.

In our example, factoring plays a crucial role when recognizing that \(2b - 4\) can be expressed as \(2(b - 2)\). By factoring the denominators, it's easier to find a common denominator and simplify the overall expression.

Some key steps in factoring include:

• Look for the greatest common factor (GCF) in all terms, and use it to factor the expression.
• Rewrite the expression using the factored form for easier manipulation in calculations.
• Practice regular factoring techniques such as grouping, using special identities, or factoring quadratics when applicable.

Factoring streamlines many algebraic operations, making it an invaluable technique for students.