Problem 16 - Algebra 2 and Trigonometry

Question

In \(3-20\) , perform the indicated additions or subtractions and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{b}{b-1}-\frac{1}{2-2 b} $$

Step-by-Step Solution

Answer

The simplest form is \( \frac{2b + 1}{2(b-1)} \), undefined for \( b = 1 \).

Step 1: Identify Common Denominators

To perform the subtraction \( \frac{b}{b-1}-\frac{1}{2-2b} \), first identify a common denominator. Notice that \( 2-2b \) can be rewritten as \( -2(b-1) \). Thus, the common denominator is \( -2(b-1) \).

Step 2: Rewrite the First Fraction

Rewrite the first fraction \( \frac{b}{b-1} \) with the common denominator \(-2(b-1)\). This is achieved by multiplying both the numerator and the denominator by \(-2\):\[ \frac{b}{b-1} = \frac{-2b}{-2(b-1)} \].

Step 3: Rewrite the Second Fraction

Rewrite the second fraction \( \frac{1}{2-2b} \) with the common denominator \(-2(b-1)\). Since it already has the denominator of \(-2(b-1)\), it remains as is:\[ \frac{1}{2-2b} = \frac{1}{-2(b-1)} \].

Step 4: Combine the Fractions

Now that both fractions have the common denominator, combine them:\[ \frac{-2b}{-2(b-1)} - \frac{1}{-2(b-1)} = \frac{-2b - 1}{-2(b-1)} \].

Step 5: Simplify the Expression

The expression simplifies to:\[ \frac{-2b - 1}{-2(b-1)} = \frac{2b + 1}{2(b-1)} \].

Step 6: Determine Undefined Values

The fraction is undefined when the denominator is zero. Set \( 2(b-1) = 0 \) to find when this occurs. Solving \( b-1 = 0 \) gives \( b = 1 \). Additionally, note that \( 2-2b = 0 \) gives \( b = 1 \) as well, confirming the undefined value.

Key Concepts

Common Denominator

When dealing with algebraic fractions, finding a common denominator is crucial to performing operations such as addition or subtraction. A common denominator is a shared term in the denominator portion of the fractions involved in the operation.

To find a common denominator, follow these steps:

• Identify the denominators involved: In our example, we have the denominators \(b - 1\) and \(2 - 2b\).
• Rewrite if necessary to reveal common factors: Notice that \(2 - 2b\) can be written as \(-2(b - 1)\). This reveals a common factor \(b - 1\) and allows us to establish \(-2(b - 1)\) as the common denominator.

By using this method, you ensure that both fractions are expressed in terms of a shared base, simplifying further operations.

Fraction Simplification

Fraction simplification helps reduce the expression to its most basic form, which makes it easier to understand and work with.

After finding a common denominator, each fraction is adjusted accordingly:

• For \(\frac{b}{b-1}\), multiply by \(-2\) to get \(\frac{-2b}{-2(b-1)}\).
• The second fraction, \(\frac{1}{2-2b}\), can be rewritten as \(\frac{1}{-2(b-1)}\), since the common denominator is already in place.

When combined, the expression:
\[ \frac{-2b}{-2(b-1)} - \frac{1}{-2(b-1)} \]
Can be simplified by combining the numerators.

This yields:
\[ \frac{-2b - 1}{-2(b-1)} \]
Simplification then results in:
\[ \frac{2b + 1}{2(b-1)} \]
Remember, simplification involves distributing, factoring, or canceling common terms where applicable.

Undefined Values

With algebraic fractions, undefined values refer to the values of the variable that result in a zero denominator. Fractions are undefined when the denominator equals zero as division by zero is not possible in mathematics.

Here's how you determine undefined values in our specific example:

• Set the common denominator \(2(b-1)\) to zero:
  Find: \( 2(b-1) = 0 \)
• Solve for \(b\):
  \( b - 1 = 0 \) leads to \( b = 1 \)

Thus, the fraction becomes undefined when \(b = 1\). Always check for undefined values when dealing with fractions to ensure the solution is valid across permissible variable ranges.