Problem 97 - A Graphical Approach to Precalculus with Limits

Question

Each inequality describes the range of average monthly temperatures \(T\) in degrees Fahrenheit at a certain location. (a) Solve the inequality. (b) Interpret the result. \(|T-61.5| \leq 12.5,\) Buenos Aires, Argentina

Step-by-Step Solution

Answer

The average temperature in Buenos Aires is between 49.0°F and 74.0°F.

Step 1: Understanding the Inequality

The inequality \(|T - 61.5| \leq 12.5\) describes all the values of \(T\) that are within 12.5 degrees of 61.5. This means \(T\) can be either more or less than 61.5 by at most 12.5.

Step 2: Splitting into Compound Inequality

The absolute value inequality \(|T - 61.5| \leq 12.5\) can be rewritten as the compound inequality \(-12.5 \leq T - 61.5 \leq 12.5\). This represents the range of \(T\) that satisfies the condition.

Step 3: Solving the Compound Inequality

To solve the compound inequality \(-12.5 \leq T - 61.5 \leq 12.5\), add 61.5 to all parts of the inequality:\(-12.5 + 61.5 \leq T \leq 12.5 + 61.5\).Simplifying both sides gives \(49.0 \leq T \leq 74.0\).

Step 4: Interpreting the Result

The solution \(49.0 \leq T \leq 74.0\) indicates that the average monthly temperature in Buenos Aires, Argentina, varies between 49.0 degrees and 74.0 degrees Fahrenheit.

Key Concepts

Understanding Absolute Value

Absolute value may sound complicated, but it's actually a simple concept. It represents the distance of a number from zero on a number line.

• For positive numbers, the absolute value is the number itself.
• For negative numbers, it's the positive version of the number.
• The notation used is vertical bars, for example, \(|x|\).

If you think about a number line, absolute value simply measures how far we are from zero, regardless of direction. In the inequality \(|T - 61.5| \leq 12.5\), the expression tells us that the distance between \(T\) and 61.5 is no more than 12.5 units away.

Solving Compound Inequality

A compound inequality involves two separate inequalities joined together. It shows us a range of possible solutions. In the exercise, the absolute value inequality \(|T-61.5| \leq 12.5\) converts into a compound inequality: \-12.5 \leq T - 61.5 \leq 12.5\.
This means \(T\) is less than or equal to 12.5 more than 61.5 and more than or equal to 12.5 less than 61.5.

• To solve, split it into two inequalities: \(-12.5 \leq T-61.5\) and \(T-61.5 \leq 12.5\).
• Add 61.5 across the board to find \(T\).

Now, we see \(-12.5 + 61.5 \leq T \leq 12.5 + 61.5\), simplifying to \(49.0 \leq T \leq 74.0\). This shows the temperature range for \(T\).

Exploring Temperature Range

The concept of temperature range explains the variability in temperature within a specific period. Here, once we've solved the inequality, we get \(49.0 \leq T \leq 74.0\).
This means that the average monthly temperatures in Buenos Aires range from 49.0 degrees to 74.0 degrees Fahrenheit.
With this understanding:

• The lower number, 49.0, represents the cooler end of the scale.
• 74.0 represents the upper limit, or the warmer end.

This range is crucial in various real-world applications, such as planning agriculture activities or tourism.

Overview of the Fahrenheit Scale

In the exercise, temperatures are measured in degrees Fahrenheit, a scale commonly used in the United States and associated regions.

Here's a simple overview of the Fahrenheit scale:

• The freezing point of water is defined as 32 degrees Fahrenheit.
• The boiling point of water is at 212 degrees Fahrenheit.
• This scale divides the interval between these points into 180 equal parts.

It's important to note that Fahrenheit temperatures are used for weather forecasting, air conditioning settings, and cooking recipes within these regions.
Understanding the scale and how it relates to daily temperatures can be helpful, especially when interpreting temperature ranges like the one in the exercise.