Problem 72 - Elementary Algebra Within Reach

Question

In Exercises 69-72, translate the phrase into an algebraic expression. Let \(x\) represent the real number. $$ \text { Nine times the difference of } h \text { and } 3 $$

Step-by-Step Solution

Answer

The algebraic expression for the given phrase is \(9*(h-3)\).

Step 1: Identify the Operations

Read the phrase carefully and identify the mathematical operations. 'Nine times' indicates multiplication, while 'difference of h and 3' informs of a subtraction operation.

Step 2: Structure the Expression

The operation of subtraction should be performed first. For this, make use of parentheses. The phrase 'the difference of h and 3' translates to '(h-3)' in algebraic form.

Step 3: Include the Multiplication

Finally, the entire phrase 'Nine times the difference of h and 3' translates to '9*(h-3)' in algebraic expression form.

Key Concepts

Algebraic Expression

An algebraic expression is a mathematical phrase that contains numbers, variables (like x or h), and operation signs. It's akin to a recipe that provides instructions on how to mix these ingredients to create a mathematical dish. Instead of using words, algebra uses symbols to represent quantities and their relationships. When translating phrases into algebraic expressions, one must understand not only the 'ingredients,' but also the correct 'method' of combining them.

For instance, the phrase given in our exercise, 'Nine times the difference of h and 3,' can seem daunting initially. But if you break it down, it simply means 9 multiplied by whatever result you get when you subtract 3 from h. Here, 'nine times' is like saying 'multiply by 9,' and 'the difference of h and 3' suggests finding what's left when you take 3 away from h. Therefore, the algebraic expression becomes 9*(h-3), a single mathematical entity that encapsulates the entire process.

Mathematical Operations

Like the basic tools in a toolbox, mathematical operations are the fundamental processes we use to manipulate numbers and variables. They include addition, subtraction, multiplication, and division. Each operation has a specific symbol that tells you what to do with the numbers or variables involved. To make things more interesting (and complicated), we also have operations like exponentiation, roots, and logarithms.

Understanding the language of these operations is crucial when translating phrases into algebraic expressions. For example, when a phrase mentions 'the sum of' something, this indicates an addition, while 'the product of' refers to multiplication. Precision in identifying these operation words within a phrase and corresponding them to their symbol is essential for constructing accurate algebraic expressions.

Order of Operations

When you have a recipe with multiple steps, it's important to follow them in the right order – or your dish might not turn out as expected. The same goes for mathematics, and this is where the order of operations comes into play. It's a set of rules that clarifies which calculations to perform first in an expression to get the correct result.

The standard order of operations is known by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Hence, when dealing with the algebraic expression 9*(h-3), you know that the operations within the parentheses (subtraction, in this case) need to be performed before the multiplication by 9. This ensures that every mathematician, student, or enthusiast will understand and solve the expression consistently, leading to the same correct result.