Problem 44 - College Algebra and Calculus: An Applied Approach

Question

Use inequality notation to describe the subset of real numbers.The price \(p\) of unleaded gasoline is not expected to go below \(\$ 2.13\) per gallon during the coming year.

Step-by-Step Solution

Answer

The inequality that describes the price range for the unleaded gasoline in the upcoming year is \(p \geq 2.13\)

Step 1: Understanding the problem

In this problem, the price \(p\) of the unleaded gasoline is not expected to drop below $2.13. Thus if \(p\) symbolizes the price per gallon, we can affirm that \(p\) is always greater than or equals to $2.13.

Step 2: Apply the inequality sign

To write this situation with an inequality sign, remember that 'greater than or equal to' is symbolized by \(\geq\). So the inequality would be \(p \geq 2.13\)

Step 3: Conclusion

The inequality \(p \geq 2.13\) mathematically communicates the given problem statement that the price of unleaded gasoline won't go below $2.13 per gallon during the coming year.

Key Concepts

Real Numbers

Real numbers are fundamental in mathematics. They include all the numbers on the number line—both rational and irrational. This means they can be fractions, decimals, or integers. Real numbers do not include imaginary or complex numbers. Each real number has a unique position on the number line making it easy to compare their sizes. They help describe quantities that we encounter every day: like temperature, distance, and yes, even price, like in our gasoline example. Here,

• Rational numbers can be exactly expressed as the ratio of two integers, like \( \frac{3}{4} \) or \( 7 \).
• Irrational numbers cannot be precisely expressed this way—common examples are \( \pi \) and the square root of numbers that aren't perfect squares.
• Integers include negative numbers, zero, and positive numbers without fractions or decimals.

Whenever you apply real numbers, you have to think about the part of the number line each number occupies. Understanding this helps with inequalities and equations.

Inequality Notation

Inequality notation is used to express the range or extent of values that a variable can take. Instead of expressing exact values, inequalities show a range. This can be defined as either:

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

In our example, we're interested in the price \( p \) of unleaded gasoline not going below \( \$2.13\). To express this, we use the 'greater than or equal to' \( \geq \) notation, resulting in \( p \geq 2.13 \). This means that \( p \), or the price, can be \( 2.13 \) or any number greater than \( 2.13 \). Inequalities are useful because they allow for ranges of solutions, which is practical in scenarios like predicting prices.

Mathematical Symbols

Mathematical symbols simplify complex ideas and allow us to convey them succinctly. They are shorthand representations that help in communicating math concepts universally. In inequalities, symbols like \( \geq \) or \( < \) denote relational concepts quickly. Here are some common ones used in inequalities:

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

By using these symbols, we can communicate phrases like "the temperature is below freezing" or "the price will not go below" efficiently. In our gasoline price example, the symbol \( \geq \) makes the statement "the price is not expected to go below \( \$2.13\)" precise and easy to understand. Symbols are not just about convenience; they form the language of all mathematical discourse, facilitating clear and unambiguous communication.