Problem 34 - Introductory Algebra for College Students

Question

Simplify complex rational expression by the method of your choice. \(\frac{\frac{1}{x-2}}{1-\frac{1}{x-2}}\)

Step-by-Step Solution

Answer

The simplified form of the given complex rational expression is \(\frac{x-2}{x-3}\).

Step 1: Recognize the Structure and Write the Reciprocal

The given expression is of the form \(\frac{a}{b}\), where \(a = \frac{1}{x-2}\) and \(b = 1 - \frac{1}{x-2}\). When dividing by a fraction, it's the same as multiplying by its reciprocal. So, the complex fraction is equivalent to \(a * \frac{1}{b}\).

Step 2: Simplify the Denominator and Reciprocal

First, simplify the denominator \(b\). Find a common denominator for \(1\) and \(\frac{1}{x-2}\). Since we notice that \(1 = \frac{x-2}{x-2}\), the denominator becomes \(b = \frac{x-2}{x-2} - \frac{1}{x-2} = \frac{x-2-1}{x-2} = \frac{x-3}{x-2}\). Now, the reciprocal of \(b\) will be \(\frac{x-2}{x-3}\).

Step 3: Multiply a and Reciprocal of b

Finally, multiply the \(a\) and reciprocal of \(b\), i.e. \(a* \left(\frac{1}{b}\right) = \frac{1}{x-2} * \frac{x-2}{x-3}\). Thus, the complex fraction simplifies and becomes \(\frac{x-2}{x-3}\).

Key Concepts

Simplifying Fractions

Simplifying fractions is like tidying up math. It means making fractions as simple as possible.
To simplify a fraction, you find the greatest common factor (GCF) of the numerator and denominator.
If the GCF is not 1, divide both by it.

• For example, to simplify \(\frac{6}{8}\), the GCF is 2.
• So, divide the numerator and denominator by 2 gives \(\frac{3}{4}\).

You can also tell a fraction is simplified when its numerator and denominator share no other common factors than 1.
Understanding simplification helps with complex fractions too.
This involves breaking them down to their simplest form, much like clearing clutter.

Rational Expressions

Rational expressions are like fractions, but with variables.
They are in the form \(\frac{p(x)}{q(x)}\), where \(p(x)\) and \(q(x)\) are polynomials.
Learning about rational expressions deepens understanding of algebra because they often appear in equations and real-world applications.

• To simplify a rational expression, factor the polynomials.
• Then, cancel out any common factors between the numerator and denominator, just as you would with numerical fractions.

A key rule is that the denominator must never be zero, as division by zero is undefined in mathematics.
This idea not only keeps math consistent but guides many problem-solving processes.
If a denominator could be zero, it's important to first find these values, known as restrictions, and keep them in mind to avoid them.

Introductory Algebra

Introductory algebra is often the first step into more complex math world.
It involves understanding how to use letters (variables) instead of numbers to solve problems.
This field sets the foundation, like learning the alphabet before reading.

• A central concept in introductory algebra is the use of equations to find unknown values.
• Equations like \(x + 5 = 10\) require finding which value of \(x\) makes the statement true.

One learns to manipulate these equations using simple operations like addition, subtraction, multiplication, and division.
This manipulation helps in operations like simplifying rational expressions.
It all aims to teach logical thinking and problem-solving skills that are essential for more advanced mathematics.