Problem 12 - Intermediate Algebra

Question

Insert the correct symbol, \(<, \leq,>,\) or \(\geq,\) in each blank. a. As many as 16 people were seriously injured: The number of people seriously injured \(\square\) 16. b. There were no fewer than 8 references to taxes in the speech: The number of tax references \(\square\) 8. c. The weight \(w\) of the roast is at most 8 pounds: \(w\) \(\square\) 8. d. The temperature \(t\) exceeded \(100^{\circ}: t \square 100\)

Step-by-Step Solution

Answer

a. \( \leq \), b. \( \geq \), c. \( \leq \), d. \( > \)

Step 1: Understand the problem statement

Read each sentence carefully to understand the context of how many or how much something is to determine the appropriate mathematical symbol to use.

Step 2: Analyze the first statement

As many as 16 people were seriously injured suggests the number cannot exceed 16. The correct symbol is \( \leq \), meaning less than or equal to.

Step 3: Analyze the second statement

No fewer than 8 references to taxes means there must be at least 8 references. The correct symbol is \( \geq \), indicating greater than or equal to.

Step 4: Analyze the third statement

At most 8 pounds means a value cannot exceed 8 pounds. The correct symbol is \( \leq \) because the weight is less than or equal to 8 pounds.

Step 5: Analyze the fourth statement

Exceeded 100 degrees implies being more than 100 degrees. Thus, the correct symbol to use is \( > \), meaning greater than.

Key Concepts

Understanding Mathematical Symbols

Mathematical symbols are essential characters that we use to express relationships or operations in mathematics. In this context, the key symbols we deal with are inequalities:

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

These symbols are not just used in isolation but help convey real-world scenarios in concise mathematical language. For example, when we say "as many as 16 people," we understand that the situation involves no more than 16 people. Thus, the right symbol is \(\leq\), meaning it is 16 or less.
Similarly, other phrases like "no fewer than" or "exceeded" dictate using specific symbols to accurately portray the mathematical concept meant by these words. Recognizing these expressions can be crucial for correctly translating statements into algebraic language.

Problem Solving with Inequalities

Solving problems with inequalities involves interpreting everyday statements and mapping them to appropriate mathematical symbols and expressions. The first step is to analyze the context given by the problem statement. Words like "at most" or "at least" help us choose the correct inequality symbol.
When tackling such problems, begin by breaking them down:

• Locate key phrases that imply a comparison, like "at most" or "exceeded," in the problem statement.
• Match these terms with inequality symbols: \(\leq\), \(\geq\), \(<\), or \(>\).
• Translate these phrases into mathematical sentences. For instance, "the weight is at most 8 pounds" becomes \(w \leq 8\).

By consistently applying this method, you can proficiently solve inequalities by expressing words in their symbolic form. The clarity in understanding can significantly enhance the efficiency of problem-solving skills.

Forming Algebraic Expressions

Algebraic expressions can include numbers, variables, and the arithmetic operations that relate them. When dealing with inequalities, expressions take on a slightly different role. They still involve numbers and variables, but they also incorporate mathematical symbols that indicate inequalities.
For example, consider the statement "the temperature \(t\) exceeded 100 degrees." To translate this into an algebraic expression, identify that it uses the term "exceeded." This suggests that \(t\) is greater than 100, represented by the expression \(t > 100\). It's crucial to understand:

• Variables, like \(t\) for temperature or \(w\) for weight, represent unknown values.
• Equations show equality while inequalities demonstrate relationships of less than or greater than.
• Forming these expressions allows us to manage and solve real-world problems mathematically.

Recognizing how complex sentences translate into algebraic language broadens our ability to articulate mathematical ideas precisely. Understanding expressions is foundational for advancing in algebra and can greatly benefit comprehensive problem-solving skills.