Problem 64 - Algebra 2

Question

Simplify each expression. \(\frac{1}{x-3}-\frac{2}{x+1}\)

Step-by-Step Solution

Answer

\( \frac{7-x}{(x-3)(x+1)} \)

Step 1: Identify the Common Denominator

To subtract two fractions, we need to find a common denominator. The denominators here are \(x-3\) and \(x+1\). Multiply these together to get the common denominator: \((x-3)(x+1)\).

Step 2: Rewrite Each Fraction

Express each fraction with the common denominator \((x-3)(x+1)\). The first fraction becomes: \[ \frac{1}{x-3} = \frac{1 \cdot (x+1)}{(x-3)(x+1)} = \frac{x+1}{(x-3)(x+1)} \] The second fraction becomes: \[ \frac{2}{x+1} = \frac{2 \cdot (x-3)}{(x+1)(x-3)} = \frac{2(x-3)}{(x-3)(x+1)} \] Note that multiplying doesn't change the value of these fractions, it only changes how they're expressed.

Step 3: Subtract the Fractions

Now that the fractions have the same denominator, we can subtract them: \[ \frac{x+1}{(x-3)(x+1)} - \frac{2(x-3)}{(x-3)(x+1)} = \frac{(x+1)-2(x-3)}{(x-3)(x+1)} \]

Step 4: Simplify the Numerator

Distribute and simplify the expression in the numerator: \[ (x+1) - 2(x-3) = x + 1 - 2x + 6 = -x + 7 \] So the expression becomes: \[ \frac{-x + 7}{(x-3)(x+1)} \]

Step 5: Final Simplified Expression

The simplified expression is: \[ \frac{7-x}{(x-3)(x+1)} \] You can leave it in this form, or write the numerator as \(-x+7\).

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in algebra that helps in making complex expressions easier to handle. It involves reducing a fraction to its simplest form, where the numerator and the denominator have no common factors other than 1. When working with algebraic expressions, this process might involve factoring out common variables or numbers and canceling them out.
An important thing to remember is that simplifying does not change the value of the fraction; it only makes it easier to understand or use in further calculations.

• Start by identifying any common factors between the numerator and the denominator.
• Divide both the numerator and the denominator by their greatest common factor.
• Simplifying can also involve canceling out terms after ensuring they don't alter the expression's value.

Reducing fractions to their simplest form often makes it much easier to perform other operations, such as addition, subtraction, multiplication, or division.

Common Denominators

When adding or subtracting fractions, finding a common denominator is crucial. The common denominator is essentially the least common multiple (LCM) of the denominators of the fractions involved. Common denominators provide a shared base for comparing, adding, or subtracting fractions directly.
For example, consider subtracting the fractions \( \frac{1}{x-3} \) and \( \frac{2}{x+1} \):

• Identify the original denominators: \( x-3 \) and \( x+1 \).
• The common denominator is their product: \((x-3)(x+1)\).
• This allows you to rewrite each fraction with a uniform base, making operations between them straightforward.

Having a common denominator simplifies the process of combining fractions by aligning the terms, enabling operations to be carried out on the numerators only.

Subtraction of Rational Expressions

Subtracting rational expressions works on similar principles to subtracting numerical fractions. Once you have a common denominator, you can subtract the numerators directly, treating them like typical algebraic expressions.
Let's consider our example:

• After obtaining a common denominator, \((x-3)(x+1)\), rewrite each fraction accordingly.
• The subtraction is performed directly on the numerators: \( (x+1) - 2(x-3) \).
• Apply algebraic operations to simplify: distribute, combine like terms, and simplify further if possible.

Remember, the process involves:

• Finding the common denominator.
• Aligning the fractions under this denominator.
• Subtracting the numerators and simplifying the resulting expression.

This method streamlines handling rational expressions and ensures accuracy in subtraction operations.

Factoring Polynomials

Factoring polynomials is a critical step in simplifying algebraic expressions, especially with rational expressions. It involves breaking down a polynomial into the product of simpler polynomials, or factors, which are sort of like the building blocks of the original expression.
Why factor? When working with fractions, factoring can help identify common factors in the numerator and denominator, making simplification possible.

• Look for any common factors in all terms of the polynomial.
• Use techniques such as grouping, difference of squares, or the quadratic formula when applicable.
• Express the polynomial as a product of these factors to make further operations easy.

In the given exercise, while the primary operation is subtraction, recognizing that factoring could simplify future steps is crucial. Factoring is a universal tool in algebra that enhances simplification, solving equations, and more efficient manipulation of expressions.