Problem 19 - Algebra 2 and Trigonometry

Question

Simplify each complex rational expression. In each case, list any values of the variables for which the fractions are not defined. \(\frac{\frac{1}{a}-1}{1-\frac{1}{a}}\)

Step-by-Step Solution

Answer

The simplified expression is \(-1\), undefined for \(a = 0\) and \(a = 1\).

Step 1: Understand the Expression

The expression given is a complex rational expression, which has a fraction in both the numerator and the denominator: \(\frac{\frac{1}{a} - 1}{1 - \frac{1}{a}}\). We want to simplify this expression.

Step 2: Find a Common Denominator

The key to simplifying a complex fraction is to find a common denominator for the fractions in both the numerator and the denominator. In this case, both the terms \(\frac{1}{a} - 1\) and \(1 - \frac{1}{a}\) have a common denominator \(a\).

Step 3: Rewrite Fractions Over a Common Denominator

Rewrite the expression so that each part of the numerator and denominator has the common denominator \(a\): - Numerator: \( \frac{1}{a} - 1 = \frac{1 - a}{a} \)- Denominator: \( 1 - \frac{1}{a} = \frac{a - 1}{a} \)

Step 4: Simplify the Complex Fraction

Using the common denominator structure, we can rewrite the original expression as a division of two fractions: \[\frac{\frac{1-a}{a}}{\frac{a-1}{a}}\]. This simplifies to: \[\frac{1-a}{a} \div \frac{a-1}{a} = \frac{1-a}{a-1}\times\frac{a}{a} = \frac{1-a}{a-1}.\]

Step 5: Simplify the Expression Further

Notice that \(1-a\) can be rewritten as \(-(a-1)\). Thus, \[\frac{1-a}{a-1} = \frac{-(a-1)}{a-1} = -1.\] The expression simplifies to \(-1\) as long as \(1-a eq 0 \rightarrow a eq 1\).

Step 6: Determine Undefined Values

For the original expression, factors like \(a-1=0\) or \(a=0\) make the denominators zero. Therefore, the values of \(a\) for which the expression is undefined are \(a = 1\) and \(a = 0\).

Key Concepts

Common Denominator

When dealing with complex rational expressions, one of the initial and crucial steps is finding a common denominator.
This process is essential because it allows us to work with equivalent fractions, making them easier to combine or compare. In the given expression, we have

• Numerator: \( \frac{1}{a} - 1 \)
• Denominator: \( 1 - \frac{1}{a} \)

Each contains fractions that share only one denominator: \( a \). Therefore, by using \( a \) as a common denominator, we can rewrite each term as

• Numerator: \( \frac{1-a}{a} \)
• Denominator: \( \frac{a-1}{a} \)

Once rewritten, it becomes possible to simplify the complex rational expression. This step forms the foundation for eliminating fractions within fractions, turning the expression into a simpler form.

Undefined Values

Understanding where an expression becomes undefined is crucial in mathematics, especially with rational expressions. For the expression \( \frac{\frac{1}{a} - 1}{1 - \frac{1}{a}} \), undefined values occur when either the original or rewritten denominator equals zero.
For the given expression:

• \( 1 - \frac{1}{a} = \frac{a-1}{a} \)

Both factors in the denominator must be looked at.

• The expression \( \frac{1-a}{a-1} \) becomes undefined when \( a - 1 = 0 \), which implies \( a = 1 \).
• Additionally, \( a = 0 \) also makes the expression undefined because of the division by zero in the rewriting steps.

Determining undefined values help us avoid these points. This ensures we maintain the integrity of mathematical operations and do not arrive at invalid solutions.

Simplification Steps

Simplifying complex rational expressions involves a clear, ordered set of steps to transform the expression into a more manageable form. Starting from our complex fraction:1. **Rewrite the expression with a common denominator.**
Both numerator and denominator of the larger fraction adopt their common form, \( \frac{1-a}{a} \) over \( \frac{a-1}{a} \).2. **Simplify using division of fractions.**
With a common denominator in place, the division of fractions now transforms into a multiplication problem: \[\frac{\frac{1-a}{a}}{\frac{a-1}{a}} = \frac{1-a}{a-1} \]3. **Further reduce by recognizing equivalent forms.**
Notice that \(1-a\) is equivalent to \(-(a-1)\). Using this identity allows us to rewrite: \[\frac{-(a-1)}{a-1} = -1\] Remember to check for cases where \(1-a eq 0\), ensuring \(a eq 1\).Going through these simplification steps with care and precision reveals that our complex expression ultimately simplifies to simply \(-1\), based on the conditions discussed earlier.