Question

A professor will cancel his sociology class if the number of students in attendance is less than or equal to \(10 .\) Which of the following expressions would give the possible values of students, \(S\), necessary for the professor to conduct class, given that \(S\) is an integer? (A) \(S<10\) (B) \(S>10\) (C) \(\quad S \leq 10\) (D) \(\quad S \geq 10\)

Short answer

The correct expression is (B) \(S > 10\).

Step-by-step solution

Understanding the Problem

The professor will cancel the class if the number of students is less than or equal to 10. We need to find the condition under which the class will not be canceled, meaning the number of students must be greater than 10.

Identifying Possible Conditions

We need to write an expression that satisfies the requirement for the class to be conducted. Since the class will be held if the number of students is greater than 10, we are looking for an expression where the values of students, \(S\), are greater than 10.

Selecting the Correct Expression

Considering all provided options: - Option (A) \(S < 10\) implies less than 10 students, leading to cancellation, thus incorrect.- Option (B) \(S > 10\) implies more than 10 students, thus the class will be held, matching our requirement.- Option (C) \(S \leq 10\) implies 10 or fewer students, leading to cancellation, thus incorrect.- Option (D) \(S \geq 10\) includes exactly 10 students, but we need more than 10 students to avoid cancellation, thus incorrect.Thus, the correct expression is \(S > 10\).

Key concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase combining numbers, variables, and arithmetic operations. In the context of our exercise, the aim is to express conditions using mathematical symbols. For instance, the inequality expressions, such as \( S > 10 \), play a pivotal role.
These expressions help us understand the number of students needed for a class to hold. Here, the variable \( S \) represents the number of students. By exploring expressions like \( S > 10 \), we use algebra to translate a real-world condition (holding the class) into a formal mathematical statement.
This translation from a spoken condition ("more than 10 students are needed") to a precise algebraic form illustrates the power of algebra. Algebraic expressions like inequalities are essential in defining boundaries or limits and facilitating decision-making in everyday scenarios.

Mathematical Reasoning

Mathematical reasoning is the logical thought process that ensures our solution is both correct and applicable. It involves understanding the problem, identifying relevant information, and making logical inferences.
For this exercise, we start by carefully interpreting the condition "less than or equal to 10" students prompting cancellation. Thus, logically, more than 10 students must attend for the class to proceed.
By breaking down each possible condition, we decide which expression fulfills the requirement. Options such as \( S < 10 \) and \( S \leq 10 \) imply cancellation conditions. Meanwhile, only \( S > 10 \) accurately represents the need for more than 10 students. This logical deduction showcases how reasoning guides us to the right solution.

Problem Solving Steps

Tackling problems like these is easier when using structured problem-solving steps. Here's a simple approach to follow:

• **Step 1: Understand the Problem.** Clearly outline what the problem is asking. In our case, we need to determine when a class is canceled based on student attendance.
• **Step 2: Identify Conditions.** Translate the problem into mathematical conditions. Here, we consider which expressions detail the requirement to conduct the class (students must be more than 10).
• **Step 3: Evaluate Options.** Examine each choice to determine which expression meets our criteria of not cancelling the class. This evaluative process rules out incorrect conditions and pinpoints \( S > 10 \) as the correct one.

Following these steps not only clarifies the problem but also increases accuracy, making complex reasoning more approachable for students.