Problem 9 - Intermediate Algebra

Question

Write each sentence using mathematical symbols. See Examples I through 4 and 6 through 8 . The product of 7 and \(x\) is less than or equal to \(-21\).

Step-by-Step Solution

Answer

The inequality is \(7x \leq -21\).

Step 1: Identify Key Terms

First, identify the mathematical operations and relationships in the sentence. "Product" refers to multiplication, "is less than or equal to" means the relationship \( \leq \), and \(x\) is our variable.

Step 2: Translate Words to Symbols

Convert the identified terms into mathematical symbols. The word "product" means we multiply 7 and \(x\), which is represented as \(7x\). The phrase "less than or equal to" is represented by \(\leq\).

Step 3: Write the Inequality

Now, write the complete inequality using the translated symbols from the previous steps. So, the sentence "The product of 7 and \(x\) is less than or equal to \(-21\)" translates to the inequality: \(7x \leq -21\).

Key Concepts

Algebraic Symbols

Understanding algebraic symbols is like learning a new language. It's an important foundation for translating verbal phrases into mathematical expressions. In algebra, symbols represent numbers and operations. For example, the symbol "\(x\)" is often used as a variable to represent an unknown value.
The arithmetic operation in the sentence, "product of 7 and \(x\)", instructs us to multiply the number 7 by this variable \(x\). That operation is written as \(7x\).
Algebraic symbols are essential for formulating expressions and equations that can then be solved or rearranged according to algebraic rules. This symbolic notation makes complex mathematical statements simpler and easier to manage.

Inequality Writing

Inequalities are a fundamental concept in mathematics, representing a range of possible values rather than a single fixed number. They help express relationships between different algebraic terms. In the sentence "is less than or equal to" is a key phrase that tells us to use the symbol \(\leq\).
This mathematical symbol represents a range of values where one expression can either be less than or equal to another. It's crucial to place the symbol correctly in the inequality to maintain the intended meaning and relationship.
So in our example, when we say "7x is less than or equal to -21", we're writing it as \(7x \leq -21\). This indicates that \(7x\) can be any value less than or even exactly equal to -21, describing a complete set of possibilities for maintaining this mathematical relationship.

Mathematical Relationships

Mathematical relationships form the backbone of understanding and interpreting word problems. They describe how variables and numbers interact with each other. In this exercise, the term "product" shows how two values interact through multiplication.
By recognizing these mathematical relationships, we translate phrases into algebraic expressions and inequalities. "The product of 7 and \(x\) is less than or equal to -21" illustrates a comparison between two expressions indicating that one side is equal to or smaller than the other.
Establishing these relationships is crucial because they help us in solving equations and inequalities accurately, helping in predicting outcomes or calculating unknown quantities.

Problem-Solving Steps

Staying methodical and disciplined in problem-solving is key to tackling mathematical exercises successfully. Here's a simple step-by-step approach:

• **Understand the Problem:** Break down the word problem into parts and identify what is being asked.
• **Identify Key Terms and Symbols:** Look for keywords indicating mathematical operations or relationships.
• **Translate Words to Symbols:** Convert the verbal description into symbols, ensuring each part of the problem is represented mathematically.
• **Formulate the Inequality or Equation:** Combine the symbols to write down the complete mathematical statement.
• **Solve and Verify:** Work through calculations to find possible solutions, and double-check to ensure accuracy.

Using these steps, we transformed "The product of 7 and \(x\) is less than or equal to -21" into \(7x \leq -21\). This systematic approach is essential for understanding and solving similar problems efficiently.