Problem 26 - Beginning and Intermediate Algebra

Question

Write each sentence as a mathematical statement. See Example 3. Negative ten is less than or equal to thirty-seven.

Step-by-Step Solution

Answer

\(-10 \leq 37\)

Step 1: Identify Key Components of the Statement

Recognize that the statement provides a relationship between two numbers: 'Negative ten' and 'thirty-seven'. Also, identify the key words 'is less than or equal to', which indicate a specific mathematical relation.

Step 2: Convert Words to Numbers and Symbols

Convert the verbal statement into a mathematical expression. 'Negative ten' translates into the number -10, 'thirty-seven' becomes 37, and 'is less than or equal to' is represented by the symbol \( \leq \).

Step 3: Combine Elements into a Mathematical Inequality

Arrange the elements to form the mathematical inequality: \(-10 \leq 37\). This inequality expresses that negative ten is less than or equal to thirty-seven.

Key Concepts

Mathematical Inequalities

Mathematical inequalities are essential tools in mathematics that allow us to compare two values. They are expressions that show relationships of greater than, less than, greater than or equal to, less than or equal to, or not equal to between different values. For example, in our exercise, "negative ten is less than or equal to thirty-seven," it can be represented as \(-10 \leq 37\). Understanding how these relationships work helps solve mathematical problems and evaluate the relative sizes of numbers.

Key terms in mathematical inequalities include:

• Less than: Symbolized by \( < \), it is used when the first value is smaller than the second.
• Less than or equal to: Represented by \( \leq \), this is used when the first value is either smaller than or equal to the second.
• Greater than: Denoted by \( > \), used when the first value is larger than the second.
• Greater than or equal to: Shown by \( \geq \), this indicates the first value is either larger than or equal to the second.
• Not equal to: Displayed as \( eq \), it shows that two values are different.

These symbols are central to understanding and using inequalities effectively.

Symbolic Representation

Symbolic representation is the process of using symbols or signs to represent numbers, operations, and relationships. By converting verbal statements into symbolic form, complex concepts become much easier to manipulate and understand.

In mathematics, particularly with inequalities, symbolic representation allows one to communicate relationships succinctly. Consider the phrase "negative ten is less than or equal to thirty-seven":

• Negative ten translates to \(-10\).
• The phrase is less than or equal to is symbolized by \( \leq \).
• Thirty-seven is simply the number \(37\).

Combining these elements helps form the symbolic expression \(-10 \leq 37\), which communicates the same idea much more concisely. Symbolic representation reduces ambiguity and is fundamental in various areas of mathematics.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and mathematical operations. They form the building blocks of more complex mathematical concepts and are crucial in representing real-world situations mathematically. Unlike inequalities, which relate quantities, algebraic expressions isolate specific quantities or data.

An example of an algebraic expression could be \( 3x + 5\), where \('x'\) is a variable that represents an unknown number.
When dealing with inequalities in algebra, we often encounter expressions combined with inequality symbols to form statements like \( 3x - 2 \leq 10\), which specifies conditions that solutions must satisfy. Inequalities using algebraic expressions are prevalent in real-world scenarios, such as budgeting or predicting outcomes, where various conditions and limitations are present. Therefore, understanding how to form and solve algebraic expressions within inequalities is a valuable skill.