Problem 112 - Introductory Algebra for College Students

Question

Write a numerical expression for each phrase. Then simplify the numerical expression by performing the given operations. The product of \(-6\) and \(-4,\) decreased by \(-5\)

Step-by-Step Solution

Answer

The simplified numerical expression is \(29\).

Step 1: Translate Verbal Phrase to Numeric Expression

Firstly, the phrase 'The product of -6 and -4' translates to \(-6 \times -4\). 'Decreased by -5' translates to minus \(-5\). Combine these two parts together, we get the expression \(-6 \times -4 -(-5)\).

Step 2: Perform the Multiplication

Next, using the order of operations, multiplication comes before subtraction. Therefore, we perform the multiplication \(-6 \times -4\) which results in \(24\). Our expression simplifies to \(24 - (-5)\).

Step 3: Perform the Subtraction

Finally, we perform the subtraction operation. As we are subtracting a negative number, this is equivalent to adding a positive number. So, \(24 -(-5)\) equals \(24+5\), which results in \(29\).

Key Concepts

Order of Operations

Understanding the order of operations in mathematics is like learning the basic rules of grammar before writing a sentence. It's essential if the sentence (or in this case, the numerical expression) is to make sense and convey the correct information. The order in which we perform arithmetic operations is crucial to obtaining the right answer.

The standard order to follow is Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This is often remembered by the acronym PEMDAS or BIDMAS (Brackets, Indices, Division and Multiplication, Addition and Subtraction).

For example, when simplifying the expression \(24 - (-5)\), we must recognize that the subtraction of a negative is effectively an addition, thus changing the expression to \(24 + 5\) before proceeding.

Arithmetic Operations

The four fundamental arithmetic operations are addition, subtraction, multiplication, and division. Each operation has its own set of rules and properties, but they all work together to help us solve numerical problems.

Multiplication, for instance, is often viewed as repeated addition, and it plays a key role in simplifying expressions when combined with negative numbers. In our example, \( -6 \times -4 \) is a multiplication of negatives, which results in a positive number, 24. Recognizing this flips the sign from negative to positive, an important concept when working with a mix of positive and negative numbers.

Translating Verbal Phrases to Numeric Expressions

When translating verbal phrases to numerical expressions, it's just like translating one language to another. Each word or phrase has a numerical counterpart that corresponds to an operation or number. For instance, 'product of' implies multiplication, while 'decreased by' indicates subtraction.

In the provided example, the phrase 'The product of -6 and -4, decreased by -5' translates to the expression \( -6 \times -4 - (-5) \) . Grasping this concept of translation is essential because it forms the foundation upon which the expression can be simplified further.

Simplifying Expressions

Simplifying expressions is the process of reducing them to their simplest form, making them easier to work with or understand. To simplify an expression, we combine like terms and apply arithmetic operations following the order of operations.

In the example \(24 - (-5)\), simplifying involves realizing that two negatives make a positive, thus \(24 - (-5)\) becomes \(24 + 5\). The final simplification leads us to 29, the simplest form of the expression. Simplification can help with understanding complex problems by breaking them down into easier, more manageable parts.