Problem 111 - 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 \(-9\) and \(-3,\) decreased by \(-2\)

Step-by-Step Solution

Answer

The simplified numerical expression for the phrase 'the product of -9 and -3, decreased by -2' is 29.

Step 1: Convert the phrase into a numerical expression

The phrase 'the product of -9 and -3, decreased by -2' can be written as: \( -9 \times -3 - (-2) \)

Step 2: Compute the product of -9 and -3

Calculate \( -9 \times -3 \) to obtain \( 27 \). This step is in accordance with order of operations, BIDMAS/BODMAS/PEDMAS, where multiplication and division comes before addition and subtraction.

Step 3: Subtract [-2] from the product

Subtract \(-(-2)\) from \( 27 \). It becomes \( 27 - [-2] = 27 + 2 = 29 \), because subtracting a negative number is equivalent to adding a positive number. This is due to the rule of 'two negatives make a positive'.

Key Concepts

Numerical Expression

A numerical expression is a mathematical phrase that combines numbers and operations. The main purpose is to denote a particular calculation or quantity. Think of it as a way to write down a problem or a sequence of calculations using numbers and operation signs like "+", "-", "×", and more.

For example, in the phrase "the product of -9 and -3, decreased by -2", the numerical expression is written as \( -9 \times -3 - (-2) \). In this expression, there are numbers involved (-9, -3, and -2) and mathematical operations (multiplication and subtraction).

Constructing a numerical expression correctly is critical for solving math problems accurately. It allows us to transform word problems or descriptive phrases into mathematical language. This gives us a clear path to follow in order to find the solution.

Multiplication

Multiplication refers to the repeated addition of the same number. In terms of negative numbers, the rules of multiplication become slightly different, but they are equally important to understand.

When you multiply two negative numbers, the result is always positive. This rule is crucial and stems from the properties of numbers and mathematical consistency.

In our problem, we have \( -9 \times -3 \), and applying the rule of multiplying two negatives, we get a positive result, which is \( 27 \).

Understanding the rule for multiplying negative numbers helps prevent errors in calculations, and is a fundamental part of algebra that will be used often.

Negative Numbers

Negative numbers are numbers less than zero, represented with a minus sign like -1, -2, or -3. They can be a bit tricky to work with, but understanding their properties makes it much easier.

A key characteristic of negative numbers is their behavior in operations:

• When you multiply or divide two negative numbers, the result is positive.
• When you multiply or divide a negative number by a positive number, the result remains negative.
• Subtracting a negative number is the same as adding its positive counterpart.

In our exercise, this is shown when we subtract \(-(-2)\). This action converts the double negative into a positive \(+2\).

Grasping these rules will enable you to handle negative numbers confidently in any math operations.

Order of Operations

Order of operations is a set of rules used to clarify which procedures to perform first in a given mathematical expression. It is often remembered by acronyms like BIDMAS, BODMAS, or PEDMAS, where:

• B or Brackets - solve expressions inside brackets first
• I or Indices/Orders - evaluate exponents or powers next
• DM or Division/Multiplication - perform from left to right
• AS or Addition/Subtraction - execute from left to right

In our solution, the multiplication \(-9 \times -3\) is performed prior to subtraction. This follows the rule where multiplication and division take precedence over addition and subtraction.

By using the order of operations correctly, you can solve mathematical expressions in the right way and ensure that complex calculations are quick and accurate.