Problem 96 - 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. \(|T-10| \leq 36,\) Chesterfield, Canada

Step-by-Step Solution

Answer

The temperature range in Chesterfield, Canada is between -26°F and 46°F.

Step 1: Understanding the Absolute Inequality

The given inequality is \(|T - 10| \leq 36\). This means that the distance between \(T\) and 10 is at most 36. In other words, \(T\) is within 36 units of 10.

Step 2: Removing Absolute Value

To solve the inequality \(|T - 10| \leq 36\), we write it as two inequalities without the absolute value: \(T - 10 \leq 36\) and \(T - 10 \geq -36\).

Step 3: Solving the First Inequality

Solve \(T - 10 \leq 36\): Add 10 to both sides: \(T \leq 46\).

Step 4: Solving the Second Inequality

Solve \(T - 10 \geq -36\): Add 10 to both sides: \(T \geq -26\).

Step 5: Combine the Solutions

Combine the inequalities: The solution for \(T\) is \(-26 \leq T \leq 46\). This shows that the average monthly temperature \(T\) in Chesterfield, Canada is between -26°F and 46°F.

Key Concepts

Inequality Solving

Inequality solving, particularly with absolute value, is slightly different from just solving regular equations. When we encounter an absolute value inequality, the key to solving it is to understand that absolute value represents the distance from zero on a number line. In this case, the inequality \( |T - 10| \leq 36\) tells us that the temperature \( T \) is within 36 units of 10.

To solve this, we need to translate the absolute value inequality into two separate inequalities. This is because the expression inside the absolute value could be either positive or negative. So, \( |T - 10| \leq 36\) becomes two inequalities:

• \( T - 10 \leq 36 \)
• \( T - 10 \geq -36 \)

This step essentially breaks down the absolute value expression, allowing us to see the range of possible values for \( T\). Solving these two inequalities separately gives us the temperature bounds for \( T \).

By adding 10 to each side of both inequalities, we get:

• \( T \leq 46 \)
• \( T \geq -26 \)

This solution explains that in Chesterfield, Canada, the potential monthly temperature range stays between these two temperatures.

Temperature Range

The concept of temperature range in the context of this problem is extremely practical, especially when interpreting data for real-world scenarios like climate or weather forecasting. When we talk about a temperature range, we are examining the limits within which a temperature is expected to fluctuate.

For the inequality solution \( -26 \leq T \leq 46 \), it indicates that the average monthly temperature in Chesterfield stays between -26°F and 46°F. This range gives us a window to expect average temperatures in that location.

Temperature ranges are important as they help in planning for

• Weather-related clothing and household preparation
• Agricultural activities that depend on temperature timings
• General human comfort and health considerations

Although extreme temperatures might sometimes appear unexpected, understanding that the monthly averages will lie within certain bounds assists in mounting appropriate responses or anticipating changes efficiently.

Mathematical Interpretation

Mathematical interpretation plays a crucial role in making sense of the results from solving inequalities. It involves translating the abstract numerical solution into a meaningful real-world context.

In our exercise, interpreting the result means comprehending what the solution \( -26 \leq T \leq 46 \) implies for Chesterfield, Canada. Mathematically, it tells us that the equilibrium of temperatures is strictly between these two values, but what does it mean practically?

By this interpretation, we understand that temperatures are likely to remain within this defined bracket most of the time. This consistency aids in anticipating seasonal changes, energy usage for heating, and various aspects of living in that region. It essentially provides a framework within which local authorities and residents plan their activities.

Thus, the process of interpreting solutions not only enhances number crunching but bonds mathematics well with everyday human activities, making it a powerful tool for societal use.