Problem 61 - Algebra for College Students

Question

Perform the indicated operations. When possible write down only the answer. $$-1\left(\frac{9-x}{2}\right)$$

Step-by-Step Solution

Answer

\(\frac{x-9}{2}\)

Step 1: Distribute the -1

Multiply \(-1\) with \(\frac{9-x}{2}\). This involves distributing \(-1\) to both terms inside the parentheses.

Step 2: Simplify the Expression

When you distribute the \(-1\), you get \(-1 \times \frac{9-x}{2} = \frac{-9+x}{2}\). Simplify this to get \(\frac{x-9}{2}\).

Key Concepts

Distributive Property

The distributive property allows you to multiply a single term across terms inside parentheses. It helps simplify expressions and solve equations. When you distribute, you multiply each term inside the parentheses by the term outside. In this case, we distribute \(-1\) across \(\frac{9-x}{2}\). This means: \-1 \times \frac{9}{2} - (-1 \times \frac{x}{2})\. Distribution simplifies problems by allowing us to break down complex expressions into simpler parts.

Notice how both terms inside the parentheses get multiplied by -1:

• \(-1 \times \frac{9}{2} = \-\frac{9}{2}\)
• \( -1 \times -\frac{x}{2} = \frac{x}{2}\)

. After distributing, you can see why \frac{-9+x}{2}\ becomes \frac{x-9}{2}.\ In essence, the distributive property makes handling complex expressions more straightforward.

Negative Exponents

Negative exponents can seem tricky, but they follow a consistent, simple rule. Any term with a negative exponent can be rewritten as the reciprocal of that term with a positive exponent. For example, \ x^{-a} = \frac{1}{x^{a}}\. This rule simplifies expressions and can make complex algebraic manipulations easier.

While this exercise didn't include explicit negative exponents, understanding them is crucial for handling algebraic expressions efficiently. Negative exponents often appear in more advanced problems. Knowing how to transform them aids in simplifying and solving algebraic equations.

Just keep in mind:

• For any non-zero number \(a\), \ a^{-1} = \frac{1}{a} \.
• This rule helps transform and reduce seemingly complex terms.
  
  

Fractions in Algebra

Handling fractions is an essential skill in algebra. The key is to treat the numerator and denominator separately while following basic arithmetic rules. Simplifying fractions involves operations like distribution and combining like terms.

In this exercise, the fraction \(\frac{9-x}{2}\) was simplified by distributing -1. The denominator remains 2 throughout, maintaining the fraction's integrity. This is an example of straightforward fraction manipulation.

Here are a few tips for working with fractions in algebra:

• Always simplify where possible. Combine like terms in the numerators or denominators.
• Use the distributive property to break down and simplify expressions.
• Keep the fractions balanced by multiplying or dividing both the numerator and denominator by the same number.

. Understanding these guidelines will make dealing with algebraic fractions much easier.

In summary, mastering fractions in algebra enhances your overall problem-solving efficiency, making you more adept at handling a variety of algebraic tasks.