Problem 16 - Fundamentals of Mathematics

Question

Translate each phrase or sentence to a mathematical expression or equation. A number increased by one.

Step-by-Step Solution

Answer

The expression is \( x + 1 \).

Step 1: Identify the Unknown

First, determine that the phrase refers to an unknown number. In algebra, we typically represent unknown numbers with a variable, often using the letter \( x \).

Step 2: Recognize the Operation

The phrase "increased by one" indicates that we need to perform an addition operation. Specifically, the original number will be added to 1.

Step 3: Construct the Mathematical Expression

Combine the variable representing the unknown number and the operation identified. So, \( x \) "increased by one" becomes \( x + 1 \).

Key Concepts

Mathematical Expression

A mathematical expression is a combination of numbers, variables, and operations that represent a particular value or relationship. These expressions can be as simple as a single number, like 2 or 7, or can become complex with multiple elements, such as variables and arithmetic operations involved.

Expressions do not include an equals sign. Thus, they differ from equations, which compare two expressions with equality. When you write or interpret mathematical expressions, you are using abstract symbols to convey a real-world meaning or problem. Learning to translate everyday language into these expressions is a fundamental skill in algebra and mathematics in general.

Variable

In math, a variable is a symbol used to represent an unknown or arbitrary number. Typically, we use letters from the alphabet for variables, with 'x' and 'y' being the most common. However, any letter can serve the purpose. Variables allow us to create expressions and equations that can adapt to different numbers and values.

• Variables are placeholders: They stand in for numbers that aren't specified or that can vary within a defined set of possibilities.
• Variables help in developing formulas and functions: By using them, we can create general rules that apply to many situations.

In our example, the phrase "a number increased by one" uses a variable, often denoted as 'x', to represent the unknown number mentioned.

Addition Operation

The addition operation is one of the basic arithmetic operations, represented by the '+' symbol. It combines two numbers into their total or sum. For example, when you add 2 and 3, you get 5.

In algebra, addition can also involve variables. If you add a number and a variable, like in the expression 'x + 1', it indicates that to find the total, you add one unit to whatever number 'x' represents.

Addition is often communicated through language cues such as 'increased by', 'plus', 'sum of', and 'added to'. Recognizing these cues is essential when translating language into mathematical expressions and equations.

Equation Translation

Translating verbal phrases or sentences into mathematical expressions or equations is a crucial step in solving algebra problems. It involves identifying keywords and concepts in the language that correspond to mathematical operations.

• Identify the unknown number: Estimate which part of the sentence refers to an unknown and represent it with a variable.
• Determine the arithmetic operation: Look for clues in the sentence, like 'increased by', which signals an addition operation.
• Formulate the expression: Combine your identified variable and the operation after fully analyzing the sentence.

In our example, "a number increased by one" translates into the expression 'x + 1', where 'x' is the variable for the unknown number and '+1' shows the addition operation.