Problem 64 - Prealgebra

Question

Write the mathematical expressions that are equivalent to each of the following English phrases. The difference of 4 and \(x\)

Step-by-Step Solution

Answer

The expression is \(4 - x\).

Step 1: Identify the Mathematical Operation

The phrase "the difference of" indicates a subtraction operation.

Step 2: Determine the Order of Terms

In the subtraction operation "the difference of 4 and \(x\)", 4 is the number from which another number \(x\) is subtracted. Thus, 4 comes first.

Step 3: Write the Expression

Using the identified operation and order, the expression is written as: \[ 4 - x \]

Key Concepts

Subtraction

Subtraction is one of the basic operations in arithmetic that involves taking one number away from another. When we talk about subtraction, it's crucial to understand both the operation itself and the meaning behind the phrase that implies it. In the example provided, the phrase "the difference of 4 and \(x\)" tells us that we are handling a subtraction problem. Here is what each part means:

• "Difference" specifically denotes subtraction. So, whenever you hear "difference," think of subtracting one quantity from another.
• The numbers involved are also important. The number mentioned first in the phrase is usually the one from which you subtract.

Therefore, in this case, 4 is being subtracted by \(x\). Understanding phrases like "less than," "decreased by," and "minus" can further enhance your ability to identify subtraction in a problem.

Order of Operations

Order of operations is fundamental when solving mathematical expressions to ensure the correct outcome. In this context, order signifies which operations to perform first when evaluating an expression. The phrase "the difference of 4 and \(x\)" indicates the need to follow the sequence as provided directly, with one clear subtraction to conduct. In general arithmetic, order of operations follows the PEMDAS rule:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

When you see "difference," it should be immediately recognized as subtraction and requires evaluating any preceding operations accordingly. Thus, knowing the order of operations is essential to avoid mistakes in multi-step problems.

Algebraic Expressions

An algebraic expression consists of numbers, variables, and operations. It represents a mathematical phrase that can include constants, like numbers, and variables, like \(x\). These expressions do not feature an equal sign, as equations do, but they can be used to set up equations. For example, the expression \(4 - x\) is an algebraic expression. Here's why:

• The constant "4" is a number that stands on its own without any accompanying variables.
• The variable "\(x\)" can represent any number; its value isn't specified in the expression.
• The operation involved is subtraction, indicated by the minus sign "-".

This type of expression is foundational in algebra and is used to model real-world situations where different quantities need to be compared or combined. Recognizing and writing algebraic expressions efficiently is key to being successful in algebra.