Problem 112 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(109-116\), 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 equals to 29.

Step 1: Transcribe the Words to Numbers and Operations

First, identify the operations outlined in the question and the numbers involved. 'The product of -6 and -4' represents the multiplication of -6 and -4. 'Decreased by -5' represents subtracting -5 from the previous result. This translates to the following numerical expression: \(-6 \times -4 - (-5)\).

Step 2: Perform the Multiplication

The phrase 'product of -6 and -4' indicates that -6 is being multiplied by -4. A product of two negative numbers is positive. So, -6 times -4 gives 24. This modifies our expression to: \(24 - (-5)\).

Step 3: Subtract the Negative Number

The phrase 'decreased by -5' essentially means adding 5 (since subtracting a negative number is equivalent to adding that number). So, our expression becomes: \(24 + 5\).

Step 4: Perform the Addition

Finally, add the numbers together. The expression simplifies to \(24 + 5 = 29\).

Key Concepts

Simplifying Numerical Expressions

When faced with numerical expressions, our goal is to 'simplify' them—this means reducing them to the simplest form possible. Simplifying often involves performing arithmetic operations in a specific order. This order is typically known as the order of operations or BIDMAS/BODMAS (Brackets, Indices/Orders, Division and Multiplication, Addition and Subtraction).

In the given exercise, the expression first involves multiplication followed by subtraction. To simplify, start with multiplication, as it precedes subtraction in the order of operations. Once multiplication is resolved, you can move on to the subtraction part. Always remember to look out for negative numbers, as they can change the result in unexpected ways. Let's see this in action with our core concepts.

Multiplication of Negative Numbers

The multiplication of negative numbers can be tricky, but it follows a simple rule: the product of two negative numbers is always positive. This is because a negative number represents the opposite of a positive number, so multiplying two 'opposites' results in a 'positive'. For example, when multiplying (-6) and (-4), think of it as taking the negative of a negative, which gives us (+24).

Why Does This Happen?

The concept of multiplying negatives can be visualized on a number line. If you move left from zero (negative direction) the same number of steps you would take right for a positive number, you are essentially reversing the reversal, landing back in the positive domain. This concept is essential because it frequently occurs in algebra and higher mathematics, so mastering it is key to mathematical proficiency.

Subtracting Negative Numbers

Subtracting negative numbers often causes confusion, but it's based on a fundamental principle: subtracting a negative is the same as adding its positive counterpart. Why? Well, when you subtract a negative, like (-5), it's akin to removing a debt; you actually gain or add to what you have. In our exercise, when we 'decrease by (-5)', we are in fact increasing the total by 5.

So, instead of complicating the expression with the subtraction of a negative, we simplify by turning it into addition: 24 - (-5) becomes 24 + 5. This transformation is a pivotal concept because it simplifies calculations and helps prevent common sign errors. It's a valuable tool in your math toolkit, particularly when dealing with complex equations or when you progress to calculus.