Problem 22 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{y+\frac{1}{y-2}}{1+\frac{1}{y-2}}$$

Step-by-Step Solution

Answer

The simplified form of the expression is \(\frac{y^2 - 2y + 1}{y - 1}\) with restrictions that it is undefined when \(y = 2\) and \(y = 1\)

Step 1: Deal with the Inner Fractions

Firstly, the inner fractions must be dealt with. It is seen that both inner fractions share the same denominator of \(y - 2\). They can therefore be written as a single fraction using the rule \(a + \frac{b}{c} = \frac{ac + b}{c}\). Applying this rule leads to: \(y + \frac{1}{y-2} = \frac{y^2 - 2y + 1}{y - 2}\) and \(1 + \frac{1}{y-2} = \frac{ y- 1}{y - 2}\)

Step 2: Simplify the Outer Fraction

Now that the inner fraction has been simplified we can substitute the resulting expressions back into the original equation. The expression simplifies to: \(\frac{\frac{y^2 - 2y + 1}{y - 2}}{\frac{ y- 1}{y - 2}}\). In this situation when a fraction is divided by another fraction, the division can be replaced by multiplication by the reciprocal of the divisor. This gives: \(\frac{y^2 - 2y + 1}{y - 2} * \frac{y - 2}{y - 1} = \frac{y^2 - 2y + 1}{y - 1}\)

Step 3: Check the Simplified Expression

The expression has been simplified to \(\frac{y^2 - 2y + 1}{y - 1}\). However, it is important to note that the original equation is undefined when \(y = 2\) and \(y = 1\). These restrictions still apply to the simplified form

Key Concepts

Fractions

Fractions are a fundamental concept in mathematics, especially in algebra. They represent a part of a whole and are written with two numbers separated by a slash, like this: \(\frac{a}{b}\), where \(a\) is the numerator (the part) and \(b\) is the denominator (the whole). When working with fractions, understanding how to manipulate them is crucial. This includes recognizing and simplifying complex fractions, which means dealing with fractions that have fractions in the numerator, denominator, or both. The objective is to reduce these complex fractions into simpler expressions that are easier to work with.

Here are some key points to remember when handling fractions:

• To combine fractions, they must have a common denominator.
• Multiply numerators and denominators separately when multiplying fractions.
• To divide fractions, multiply by the reciprocal of the divisor.
• Simplification involves breaking down fractions into their simplest form by finding the greatest common factor of the numerator and denominator.

Understanding fractions can provide a solid foundation for more advanced mathematical concepts.

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves restructuring an expression in a way that's easier to work with. The goal is to make calculations simpler or to solve equations more efficiently. In the context of rational expressions, simplification often means converting complex fractions into a simpler form.

In algebra, simplification may involve:

• Combining like terms: Terms in an expression that have the same variables raised to the same power.
• Applying the distributive property to expand or factor expressions.
• Cancelling common factors in fractions.

For example, in the problem at hand, complex fractions were involved. The process included rewriting the expression by dealing with the inner fractions first, then simplifying the entire expression by using algebraic rules.

Pay attention to restrictions such as undefined values when variables make the denominator zero. These are crucial for ensuring expressions are valid.

Prealgebra

Prealgebra forms the bedrock of all algebraic understanding, essential for progressing into more complex topics. It begins by introducing basic operations and concepts, setting the stage for high school algebra and beyond.

The fundamental topics in prealgebra include:

• Arithmetic of natural numbers, integers, and fractions.
• Introduction to variables and simple equations.
• Basic properties of operations: associative, distributive, and commutative.
• The concept of factors and multiples.

The exercise provided operates on concepts from prealgebra like dealing with fractions and variables, and applying these to simplify an algebraic expression. Understanding these basics is crucial before tackling more advanced algebraic concepts.

Mathematical Operations

Mathematical operations such as addition, subtraction, multiplication, and division form the core of arithmetic and algebra. They are used to manipulate numbers and expressions to eventually simplify or solve them. When handling rational expressions, these operations become the tools with which expressions are rewritten or broken down.

Important concepts include:

• Order of operations: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (PEMDAS).
• Multiplication by the reciprocal when dividing fractions, as seen in the solution process.
• Integration of different operations to simplify expressions.

These operations are applied systematically to ensure accuracy in simplification and solving. Understanding and mastering these operations can significantly ease the way we deal with algebraic expressions like the one in this exercise.