Problem 58 - Algebra 1

Question

EQUATIONS AND INEQUALITIES Match the verbal sentence with its mathematical representation. The difference of \(x\) and 16 is 32

Step-by-Step Solution

Answer

The mathematical representation of the verbal sentence 'The difference of \(x\) and 16 is 32' is \(x - 16 = 32\).

Step 1: Understanding the Phrase

In the phrase, 'the difference of \(x\) and 16 is 32', 'the difference of \(x\) and 16' translates to \(x - 16\). 'Is' in mathematics can serve as the equals sign, '='. Lastly, '32' directly translates to 32.

Step 2: Formulating the Equation

Taking the understood parts and assembling them together forms an equation. Doing so translates 'the difference of \(x\) and 16 is 32' to \(x - 16 = 32\).

Key Concepts

Inequalities

Inequalities are an essential part of mathematics, offering a way to express relationships between expressions that are not exactly equal. Unlike equations, where both sides are equivalent, inequalities show that one side can be larger or smaller than the other. Think of them as comparisons that help us understand how quantities relate to one another.
Inequalities are represented by special symbols:

• < : less than
• > : greater than
• \( \leq \): less than or equal to
• \( \geq \): greater than or equal to

Graphing inequalities is another way to visualize solutions. You might use a number line, where open or closed circles denote whether endpoints are included. As you get comfortable with inequalities, they become valuable tools for representing and solving everyday problems.

Mathematical Representation

Mathematical representation allows us to take real-world situations and put them into a form that we can manipulate and analyze. This is often done by using letters to stand for numbers we do not know yet, which we call variables. By using symbols and expressions instead of words, it becomes easier to see relationships and solve problems.
The power of mathematical representation comes from its ability to simplify complex ideas. Instead of explaining a situation with many words, you can use an expression, equation, or inequality to communicate the same idea clearly and concisely. For example, in the given exercise, "the difference of \(x\) and 16" is represented mathematically as \(x - 16\). This simple notation helps us solve the problem systematically.

Verbal Sentences

Verbal sentences in math convert the language we speak every day into phrases that have specific mathematical meanings. Understanding these sentences is crucial because they are often the first step in formulating equations or inequalities.
Consider the sentence "the difference of \(x\) and 16 is 32." Each part of the sentence gives us clues:

• "The difference of \(x\) and 16" suggests a subtraction operation: \(x - 16\).
• "Is" translates to an equals sign \(=\) in mathematical language.
• "32" remains a constant number.

Breaking down verbal sentences this way helps us identify the correct mathematical operations and representations needed for problem-solving.

Equation Translation

Equation translation is the process of converting verbal phrases or real-world scenarios into mathematical equations. Successfully translating sentences into equations is a key skill in algebra and problem-solving.
The exercise demonstrates this process clearly. Given the verbal sentence "the difference of \(x\) and 16 is 32," we translate each part step-by-step:

• "The difference of \(x\) and 16" becomes \(x - 16\).
• "Is" transforms into the equals sign \(=\).
• "32" remains 32.

Thus, we form the equation \(x - 16 = 32\). This skill allows us to move from description to calculation, fundamentally bridging communication and computation.