Simplify (1-1/(x+2))/(1+3/(x-2))
The problem provided is asking for the simplification of a complex rational expression. It presents a fraction where the numerator is another fraction subtracted from 1, specifically (1 - 1/(x+2)), and the denominator consists of 1 plus a different fraction, specifically 3/(x-2). The goal is to manipulate and simplify this expression so that it is written in a more streamlined form, possibly as a single fraction or another simplified mathematical expression. This typically involves combining the terms and fractions through common denominators, factoring, and canceling out terms if possible to achieve the simplest form.
$\frac{1 - \frac{1}{x + 2}}{1 + \frac{3}{x - 2}}$
Answer
Solution:
Simplifying the Complex Fraction
Step 1: Multiply by the Common Denominator
To eliminate the complex fraction, we multiply both the numerator and denominator by $(x+2)(x-2)$.
Step 1.1: Apply Multiplication
Multiply the entire expression by $\frac{(x+2)(x-2)}{(x+2)(x-2)}$ to get:
$$\frac{(x+2)(x-2)}{(x+2)(x-2)} \cdot \frac{1 - \frac{1}{x+2}}{1 + \frac{3}{x-2}}$$
Step 1.2: Combine Terms
Combine the terms to simplify the expression:
$$\frac{(x+2)(x-2) \cdot \left(1 - \frac{1}{x+2}\right)}{(x+2)(x-2) \cdot \left(1 + \frac{3}{x-2}\right)}$$
Step 2: Distribute the Multiplication
Apply the distributive property to both the numerator and denominator:
$$\frac{(x+2)(x-2) \cdot 1 - (x+2)(x-2) \cdot \frac{1}{x+2}}{(x+2)(x-2) \cdot 1 + (x+2)(x-2) \cdot \frac{3}{x-2}}$$
Step 3: Simplify by Canceling Common Factors
Step 3.1: Cancel $(x+2)$
Cancel the $(x+2)$ term in the numerator:
$$\frac{(x+2)(x-2) \cdot 1 - \cancel{(x+2)}(x-2) \cdot \frac{1}{\cancel{x+2}}}{(x+2)(x-2) \cdot 1 + (x+2)(x-2) \cdot \frac{3}{x-2}}$$
Step 3.2: Cancel $(x-2)$
Cancel the $(x-2)$ term in the denominator:
$$\frac{(x+2)(x-2) \cdot 1 - (x-2) \cdot -1}{(x+2)\cancel{(x-2)} \cdot 1 + (x+2)\cancel{(x-2)} \cdot 3}$$
Step 4: Simplify the Numerator
Step 4.1: Expand and Combine Like Terms
Expand the terms in the numerator and combine like terms:
$$\frac{(x+2)(x-2) + (x-2) \cdot -1}{(x+2)(x-2) + (x+2) \cdot 3}$$
Step 5: Simplify the Denominator
Step 5.1: Expand and Combine Like Terms
Expand the terms in the denominator and combine like terms:
$$\frac{(x-2)(x+1)}{(x+2)(x-2 + 3)}$$
Step 6: Cancel the Common Factor $(x+1)$
Step 6.1: Cancel the Common Factor
Cancel the $(x+1)$ term in both the numerator and denominator:
$$\frac{\cancel{(x-2)}\cancel{(x+1)}}{(x+2)\cancel{(x+1)}}$$
Step 6.2: Final Simplified Expression
The final simplified expression is:
$$\frac{x-2}{x+2}$$
Knowledge Notes:
To simplify a complex fraction, we often multiply the numerator and denominator by the least common denominator (LCD) of all the fractions within the complex fraction. This process eliminates the smaller fractions and allows us to work with a single, simplified fraction.
In this case, the LCD is the product of the distinct linear factors present in the denominators of the smaller fractions, which are $(x+2)$ and $(x-2)$. Multiplying by the LCD helps to clear the fractions within the complex fraction.
The distributive property is used to multiply terms within parentheses, and it is essential for combining like terms and simplifying expressions.
When simplifying expressions, always look for common factors that can be canceled out. This is particularly useful when dealing with rational expressions, as it simplifies the fraction to its lowest terms.
The final step is to ensure that the expression is fully simplified by canceling out any common factors in the numerator and denominator, resulting in the simplest form of the expression.
