Problem 8 - Elementary and Intermediate Algebra

Question

Simplify each expression. a. \(\frac{(x+2)(x-2)}{(x+1)(x+2)}\) b. \(\frac{y(y-2)}{9(2-y)}\) c. \(\frac{(2 m+7)(m-5)}{(2 m+7)}\) d. \(\frac{x \cdot x}{x \cdot x(x-30)}\)

Step-by-Step Solution

Answer

a. \(\frac{x-2}{x+1}\), b. \(-y\), c. \(m-5\), d. \(\frac{1}{x-30}\).

Step 1: Simplify Part (a)

Given the expression \(\frac{(x+2)(x-2)}{(x+1)(x+2)}\). First, notice that the term \((x+2)\) appears in both the numerator and the denominator. Since it is a common factor, we can simplify by canceling it out.The expression then simplifies to \(\frac{x-2}{x+1}\).

Step 2: Simplify Part (b)

For the expression \(\frac{y(y-2)}{9(2-y)}\), recognize that \(2-y\) can be written as \(-(y-2)\) because \(2-y = -(y-2)\).Thus, we can express the denominator as \(-1\cdot(y-2)\). Now the expression is \(\frac{y(y-2)}{-1 \cdot (y-2)}\).Cancel the \((y-2)\) factor in both the numerator and the denominator, leaving us with \(-\frac{y}{1} = -y\).

Step 3: Simplify Part (c)

Given \(\frac{(2 m+7)(m-5)}{(2 m+7)}\), notice that \((2 m+7)\) exists in both the numerator and denominator. Cancel the \((2 m+7)\) term.The expression simplifies to \(m-5\).

Step 4: Simplify Part (d)

Considering the expression \(\frac{x \cdot x}{x \cdot x \cdot (x-30)}\), note that there are two factors of \(x\) in both the numerator and the denominator.Cancel two \(x\) terms from both, which simplifies the expression to \(\frac{1}{x-30}\).

Key Concepts

Simplifying Fractions

Simplifying fractions makes them easier to understand and work with. This process involves reducing the fraction to its simplest form by eliminating factors that are common to both the numerator and the denominator. To simplify a fraction, follow these steps:

• Identify any common factors in both the numerator and the denominator.
• Divide both the numerator and the denominator by their greatest common factor.
• Rewrite the fraction using the reduced terms.

For example, consider the expression \(\frac{(x+2)(x-2)}{(x+1)(x+2)}\). Here, \((x+2)\) is a common factor. By canceling \((x+2)\), the fraction simplifies to \(\frac{x-2}{x+1}\). This is a cleaner and more manageable expression.
Understanding how to simplify fractions helps in solving math problems efficiently and accurately.

Canceling Common Factors

Canceling common factors is a crucial part of simplifying fractions. When common factors appear in the numerator and denominator of a fraction, they "cancel out". This means you can divide both by the same factor without changing the value of the expression.
Consider the exercise \(\frac{(2m+7)(m-5)}{(2m+7)}\) from earlier. Here, the term \((2m+7)\) is present in both the numerator and denominator. By canceling it, we simplify the fraction to just \(m-5\). This method requires you to:

• Identify common elements that appear as factors in both parts of the fraction.
• Ensure these elements are multiplied in both positions, allowing them to be eliminated through division.

Remember, the goal is to streamline equations for easier computation and better understanding.

Negative Signs in Algebra

Negative signs in algebra can sometimes create confusion, especially when they are part of a larger expression or fraction. It's important to properly handle negative signs to prevent errors and make algebraic manipulation smoother.
When simplifying fractions like \(\frac{y(y-2)}{9(2-y)}\), notice that \(2-y\) is equivalent to \(-(y-2)\), because reversing the order of subtraction introduces a negative. This insight allows us to rewrite the expression as \(\frac{y(y-2)}{-1 \cdot (y-2)}\). By canceling the \((y-2)\) factor, we end up with \(-y\).

• Recognize when reversing terms requires adjusting the sign.
• Remember that a negative sign can be factored out or distributed, influencing the entire expression.
• For any two expressions \(a\) and \(b\), \(a-b=-(b-a)\).

Being comfortable with negative signs helps you solve algebra problems more confidently and accurately.