Problem 115 - 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 difference between \(-6\) and the quotient of 12 and \(-4\)

Step-by-Step Solution

Answer

The simplified numerical expression is -3.

Step 1: Write the Numerical Expression

The numerical expression for the phrase will be: \(-6 - (12/-4)\). 'The difference between -6 and..' means we are subtracting from -6. 'the quotient of 12 and -4' indicates we are dividing 12 by -4.

Step 2: Perform Division

First, carry out the division operation inside the brackets which is 12 divided by -4, giving -3. So the expression becomes: \(-6 - (-3)\). According to the order of operations, division comes before subtraction.

Step 3: Perform Subtraction

Next, subtract -3 from -6. The subtraction of a negative number is same as the addition of a positive number. So the final simplified numerical expression becomes: \(-6 + 3 = -3\).

Key Concepts

Order of Operations

Understanding the order of operations is crucial when simplifying numerical expressions. It's like giving directions to someone; if the steps are out of order, the destination may not be reached correctly. The order of operations, often remembered by the acronym PEMDAS, directs us to perform calculations in this order: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

For example, when we see an expression like \( -6 - (12 / -4) \), we must first address any calculations inside parentheses. This guidance ensures that we perform the division \( 12 / -4 \) before we move on to subtract from -6. By respecting this order, we prevent potential mistakes and arrive at the correct simplified expression.

Arithmetic Operations

Arithmetic operations form the basis of all mathematical calculations. The four primary operations are addition, subtraction, multiplication, and division. Each operation has a specific symbol and rule set governing its use. To simplify numerical expressions accurately, students must be proficient with these operations.

Take our expression \( -6 - (12 / -4) \). Here, we encounter both division and subtraction. Division is indicated by the symbol '/' and is the process of finding out how many times one number is contained within another. Upon executing the division \(12 / -4\), which equals \( -3 \), we then move to subtraction – indicated by the symbol '-'. Subtraction tells us to take away one quantity from another; in this case, we are taking \( -3 \), the result of the division, away from \( -6 \). Recognizing and applying these operations correctly is a key skill in mathematics.

Mathematical Expressions

Mathematical expressions are phrases that contain numbers, operations, and sometimes variables that represent a particular value. Simplifying such expressions is akin to translating a complex sentence into a simple one while maintaining its original meaning. In our given problem, the expression \( -6 - (12 / -4) \) represents the actions to be taken to find the solution. The numerical expression conveys a mathematical idea in a compact form that can be worked through systematically.

Oftentimes, the complexity increases with the addition of variables, exponents, and multiple operations, but the basic principle remains: express and then simplify using the appropriate rules of arithmetic and the order of operations. Being adept at interpreting and simplifying expressions is an essential skill, helping students navigate through more advanced levels of mathematics with confidence.