Problem 21 - Fundamentals of Mathematics

Question

Translate each phrase or sentence to a mathematical expression or equation. A number minus the opposite of five.

Step-by-Step Solution

Answer

The expression is \( x + 5 \).

Step 1: Identify the Key Terms

First, we need to translate the spoken phrase into mathematical components. The phrase 'a number' is typically represented by the variable, which is commonly denoted as \( x \). The term 'the opposite of five' means \(-5\) because 'opposite' in mathematics refers to changing the sign.

Step 2: Structure the Expression

Now, we need to structure these components into the mathematical expression based on the operation mentioned. The phrase 'minus' indicates a subtraction operation. Therefore, 'a number minus the opposite of five' translates to \( x - (-5) \).

Step 3: Simplify the Expression

Finally, simplify the expression \( x - (-5) \). Remember that subtracting a negative number is equivalent to adding the positive version of that number. Therefore, \( x - (-5) \) simplifies to \( x + 5 \).

Key Concepts

Opposite of a Number

In mathematics, understanding the concept of the "opposite of a number" is fundamental. Put simply, the opposite of a number refers to a value that, when added to the original number, equals zero. This is also known as the additive inverse. For example, the opposite of 5 is \( -5 \) because \( 5 + (-5) = 0 \).- When confronted with phrases like "the opposite of five," interpret this as changing the sign of five.- It doesn't matter whether the original number is positive or negative; you simply flip the sign.Recognizing the opposite is crucial in translating verbal expressions to algebraic ones, because it dictates how we structure the mathematical operations involved. For example, in our given exercise, identifying "the opposite of five" as \( -5 \) allows us to correctly translate the phrase into a mathematical expression.

Variable Representation

A major element of algebra is the representation of unknown quantities through variables. The term "a number" in verbal expressions is typically reflected as a variable in equations. The most common choice for a variable is \( x \), but you can use any letter or symbol that makes sense given the context. - It acts as a placeholder for an unknown or changeable value.- Using variables helps us work with mathematical operations with flexibility and precision.In our example exercise, "a number minus the opposite of five," the ambiguous 'a number' is efficiently represented as \( x \), enabling us to proceed with defining the other components of the expression.

Mathematical Operations

Grasping mathematical operations is key to building expressions from verbal phrases. Let's look at how operations like subtraction are used:- **Subtraction:** In algebra, the operation indicated by the word "minus" is subtraction.When translating an expression such as "a number minus the opposite of five," we need to understand:1. The "minus" cues subtraction.2. The "opposite of five" is \( -5 \). Bringing these together, the phrase corresponds to: \( x - (-5) \).- **Simplification:** When we subtract a negative number, we need to remember the rule: subtracting a negative is the same as adding the positive.- Thus \( x - (-5) \) simplifies to \( x + 5 \).This simplification is essential for correctly solving and understanding algebraic expressions. Recognizing how these operations work is crucial to the translation from verbal phrases to mathematical equations.