Problem 113 - Prealgebra and Introductory Algebra

Question

State whether the expression is equivalent to \(\frac{a}{b}\) or \(-\frac{a}{b}\) Assume that \(a\) and \(b\) are nonzero integers. $$-\frac{-a}{b}$$

Step-by-Step Solution

Answer

The expression \(-(-a/b)\) is equivalent to \(a/b\).

Step 1: Interpreting the expression

In the expression \(-(-a/b)\), there is a negative sign outside the brackets, and another one in front of \(a\) which is the numerator. It's important to point out that two negatives combined form a positive. This is based on the rules of sign in basic algebra.

Step 2: Simplifying the expression

After making the observation from Step 1, we can simplify \(-(-a/b)\) as follows: From the rules of algebra concerning negative numbers, the expression simplifies to \(a/b\). The two negatives signs combine to form a positive, meaning that \(-(-a)\) just becomes \(a\), hence the expression \(-(-a/b)\) becomes \(a/b\).

Step 3: Outlining the equivalence

Now we can state confidently that the expression \(-(-a/b)\) is equivalent to \(a/b\), not \(-a/b\). This conclusion was reached by simplifying \(-(-a/b)\), and observing that the simplified expression is \(a/b\).

Key Concepts

Simplifying Expressions

Let's start by understanding the concept of simplifying expressions in algebra. Simplifying an expression means to rewrite it in the most reduced and concise form possible. This involves combining like terms and performing arithmetic operations that produce an equivalent, simpler expression.
Consider the expression \(-(-a/b)\). To simplify it, we must carefully apply the rules of arithmetic and algebra. The expression involves two negative signs. According to the rules of mathematics, two negatives cancel each other out and turn into a positive.
So, when you see a negative sign outside the negative of a fraction, like in \(-(-a/b)\), you realize this is just \(+a/b\). The negative sign in front of \(-a\) changes \(-a\) to \(+a\), resulting in the simplified expression \(+a/b\). It's essential to simplify expressions to facilitate easier manipulation and to compare such forms.

Negative Numbers

Negative numbers are one of the basic elements in algebra. They represent values less than zero and are denoted by a minus sign (-). Understanding how negative numbers interact with each other is crucial.
Here are some key points to remember:

• If you multiply or divide two negative numbers, you get a positive number.
• If you add or subtract a negative number with a positive number, it decreases the positive number.
• When you have two negatives, like \(-(-x)\), it becomes positive because the negatives cancel out.

In our exercise, we saw \(-(-a/b)\). It involves two negative signs before the numerator, and according to the rules, this results in a positive fraction. Recognizing these sign rules will help you not only in simplifying expressions but also in solving equations and inequalities.

Basic Algebra

Basic Algebra is the foundation of more advanced mathematical studies and is integral for problem-solving. It involves using symbols, typically letters, to represent unknown values in mathematical expressions and equations.
In algebra, expressions allow us to write mathematical language flexibly without knowing the actual values at play. In the given expression, \(-(-a/b)\), algebra helps clarify operations involving variables like \a\ and \b\.

• We use rules, such as sign rules, to simplify these expressions.
• It emphasizes the application of operations on expressions without knowing the specific numbers.

Developing an understanding of basic algebra involves getting comfortable with variables and learning how to manipulate them. Skills in basic algebra are essential for further studies in mathematics, enabling one to solve equations and understand functions effectively.