Problem 86 - A Graphical Approach to College Algebra

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. \(|T-43.5| \leq 8.5,\) Punta Arenas, Chile

Step-by-Step Solution

Answer

The average temperature is between 35°F and 52°F.

Step 1: Understand the Absolute Value Inequality

The given inequality is \(|T - 43.5| \leq 8.5\). This inequality states that the difference between \(T\) and 43.5 is at most 8.5. To solve it, we must express it as a compound inequality without absolute value.

Step 2: Convert to Compound Inequality

The inequality \(|T - 43.5| \leq 8.5\) can be rewritten as \(-8.5 \leq T - 43.5 \leq 8.5\). This compound inequality shows that \(T\) is no more than 8.5 units away from 43.5 in either direction.

Step 3: Solve the Compound Inequality

To find the range of \(T\), we solve \(-8.5 \leq T - 43.5\) and \(T - 43.5 \leq 8.5\).- For \(-8.5 \leq T - 43.5\), add 43.5 to both sides to get \(35 \leq T\).- For \(T - 43.5 \leq 8.5\), add 43.5 to both sides to get \(T \leq 52\).

Step 4: Combine the Solutions

Combine the two inequalities from the previous step to express as one range: \(35 \leq T \leq 52\). This indicates that the average monthly temperature \(T\) is between 35 degrees and 52 degrees Fahrenheit.

Step 5: Interpret the Result

The average monthly temperature in Punta Arenas, Chile, ranges from 35 degrees Fahrenheit to 52 degrees Fahrenheit. This range indicates relatively mild temperatures, likely due to its southern location and oceanic influence.

Key Concepts

Compound Inequalities

Absolute value inequalities often lead to compound inequalities. This is because the absolute value symbol \( | \) indicates a range of values. The expression \( |T - 43.5| \leq 8.5 \) can be understood as asking how far the temperature \( T \) is from 43.5 degrees. In this case, it means \( T \) can be up to 8.5 units away, either higher or lower.
To convert it into a compound inequality, we need to consider both these directions. We split it into two parts:

• \( T - 43.5 \leq 8.5 \)
• \( T - 43.5 \geq -8.5 \)

These parts combine to form a compound inequality: \(-8.5 \leq T - 43.5 \leq 8.5\)
Thus, compound inequalities help describe data that fits within a specific range on a number line, making them quite useful for real-world applications.

Solving Inequalities

Solving these compound inequalities involves a few simple steps. Let's break down the steps to solve the inequality \(-8.5 \leq T - 43.5 \leq 8.5\):

• First, add 43.5 to the left side of the inequality: \( -8.5 + 43.5 \leq T \) resulting in \( 35 \leq T \).
• Next, do the same on the right side: \( T \leq 8.5 + 43.5 \), simplifying to \( T \leq 52 \).

This process gives us the range \( 35 \leq T \leq 52 \).
By solving inequalities, you find a set of possible solutions or values that satisfy certain conditions. This method is essential for determining the limits within which real-world variables, like temperature, can fluctuate.

Temperature Range Interpretation

When you obtain a solution like \( 35 \leq T \leq 52 \), it's helpful to interpret it in terms of real-world context, here focusing on Punta Arenas, Chile. This solution tells us that the average monthly temperature varies between 35 and 52 degrees Fahrenheit.
This range suggests relatively mild temperatures compared to more extreme environments. The southern location and ocean's moderating effects contribute to this stability. These insights can inform various decisions, such as when deciding the best time to visit or understanding local weather patterns.
By interpreting the range, students can connect mathematical conclusions to everyday implications, strengthening their understanding of both mathematics and geography.