Problem 44 - Algebra A Combined Function

Question

Use the distributive property to write each expression without parentheses. Then simplify the result, if possible. See Examples 7 through 12. $$ -\frac{1}{2}(2 r+11) $$

Step-by-Step Solution

Answer

-r - \frac{11}{2}

Step 1: Identify the Distributive Property

The distributive property states that for all numbers \(a\), \(b\), and \(c\), \(a(b+c) = ab + ac\). Here, the expression is \(-\frac{1}{2}(2r+11)\), where \(-\frac{1}{2}\) is distributed over \(2r\) and \(11\).

Step 2: Distribute \(-\frac{1}{2}\) to Each Term

Apply the distributive property: Multiply \(-\frac{1}{2}\) by each term inside the parentheses. Calculate \(-\frac{1}{2} \times 2r\) and \(-\frac{1}{2} \times 11\).

Step 3: Perform the Multiplication

Calculate \(-\frac{1}{2} \times 2r = -\frac{1}{2} \times 2 \times r = -r\). Calculate \(-\frac{1}{2} \times 11 = -\frac{11}{2}\).

Step 4: Write the Result as a Simplified Expression

Combine the results from Step 3: \(-r - \frac{11}{2}\).This is the simplified expression without parentheses.

Key Concepts

Simplifying Algebraic Expressions

Simplifying algebraic expressions involves reducing the expression to its simplest form. This process often includes removing parentheses, combining like terms, and reducing fractions. To start, identify any terms that can be combined or operations that can be performed. This typically involves using mathematical properties such as the distributive property.

• The distributive property allows you to multiply a term outside the parenthesis by each term inside the parenthesis separately.
• After distributing, combine any like terms to simplify further if necessary.

In our example, applying the distributive property allowed us to remove the parentheses and simplify the expression into \(-r - \frac{11}{2}\). It's crucial to perform each step carefully to ensure accuracy and avoid mistakes in the simplification process.

Multiplication of Rational Numbers

Understanding rational numbers is key when working with algebraic expressions. Rational numbers include any number that can be expressed as a fraction, such as integers, fractions, and decimals. The multiplication of rational numbers involves multiplying their numerators and denominators separately.

• First, make sure all terms are in fraction form if possible.
• Multiply the numerators together to find the new numerator.
• Multiply the denominators together to find the new denominator.
• Always simplify the resulting fraction to its simplest form.

In the exercise, multiplying \(-\frac{1}{2} \times 2r\) resulted in \(-r\), and multiplying \(-\frac{1}{2} \times 11\) resulted in \(-\frac{11}{2}\). These operations illustrate the straightforward yet crucial role of multiplication when handling rational numbers.

Properties of Operations in Algebra

Algebra relies on several key properties that make solving equations and simplifying expressions possible. These properties include:

• Distributive Property: This allows you to multiply a single term by each term within a set of parentheses, as shown in \(a(b+c) = ab + ac\).
• Associative Property: It states that how you group numbers in addition or multiplication does not affect the result, i.e., \((a+b)+c = a+(b+c)\).
• Commutative Property: The order in which you add or multiply numbers doesn't matter, such as \(a+b = b+a\).

Understanding and applying these fundamental properties properly can significantly simplify solving algebra problems. In the provided exercise, the distributive property was used to distribute \(-\frac{1}{2}\) across terms \(2r\) and \(11\), demonstrating its power to remove parentheses and simplify expressions efficiently.