Problem 100 - College Algebra

Question

The inequality \(|T-57| \leq 7\) describes the range of monthly average temperature, \(T,\) in degrees Fahrenheit, for San Francisco, California. The inequality \(|T-50| \leq 22\) describes the range of monthly average temperature, \(T\), in degrees Fahrenheit, for Albany, New York. Solve each inequality and interpret the solution. Then describe at least three differences between the monthly average temperatures for the two cities.

Step-by-Step Solution

Answer

The range of monthly average temperatures for San Francisco is [50, 64] degrees Fahrenheit, and for Albany, it is [28, 72] degrees Fahrenheit. Albany has both hotter summers and colder winters compared to San Francisco, suggesting it experiences a larger temperature range over the year.

Step 1: Solve the inequalities for San Francisco

To solve \(|T-57| \leq 7\) consider two cases: T - 57 ≤ 7 when T ≥ 57 and -(T - 57) ≤ 7 when T < 57. Solving these inequalities gives T ≤ 64 and T ≥ 50, respectively. Therefore, the solution set for \(T\) is [50, 64].

Step 2: Solve the inequalities for Albany

To solve \(|T-50| \leq 22\) consider two cases: T - 50 ≤ 22 when T ≥ 50 and -(T - 50) ≤ 22 when T < 50. Solving these inequalities gives T ≤ 72 and T ≥ 28, respectively. Therefore, the solution set for \(T\) is [28, 72].

Step 3: Interpret the solutions

The solutions imply that the range of monthly average temperatures for San Francisco is between 50 and 64 degrees Fahrenheit, whereas for Albany, it ranges between 28 and 72 degrees Fahrenheit. This shows the differences in climates of the two cities.

Step 4: List at least three differences

1. Albany has a wider range of monthly average temperatures than San Francisco. 2. The minimum average temperature for Albany is much lower than San Francisco's. 3. The maximum average temperature for Albany is higher than San Francisco's. These reflect the different climates with Albany experiencing both hotter summers and colder winters.

Key Concepts

Understanding Absolute Value Inequalities

Absolute value inequalities, like \(|T - 57| \leq 7\), are equations where the absolute value of an expression is set to be less than or equal to a number. To solve them, consider two scenarios because absolute values can result from both positive and negative input.

• When the expression inside the absolute value is positive or zero.
• When the expression inside the absolute value is negative.

For \( |T - 57| \leq 7\), break it into two cases:

• Case 1: \(T - 57 \leq 7\) \Rightarrow \(T \leq 64\)
• Case 2: \(- (T - 57) \leq 7\) \Rightarrow \(T \geq 50\)

Thus, the temperature \(T\) for San Francisco is between \[50, 64\] degrees Fahrenheit.
To master absolute value inequalities:- **Understand** that the solution is not just about equality, but also about considering both possible scenarios of the expression inside the absolute value.- **Practice** with various inequalities to become confident in solving and interpreting them.

Steps for Solving Inequalities

Solving inequalities often requires addressing multiple cases, especially when absolute values are involved. Follow these key steps:

• **Isolate** the absolute value expression if necessary, making it the subject of your equation.
• **Consider two cases**: one where the inside of the absolute value is positive, and another where it is negative.
• **Solve** each case separately to find the range of solutions.

For Albany's inequality \( |T - 50| \leq 22\), follow the steps:

• Case 1: \(T - 50 \leq 22\) leads to \(T \leq 72\)
• Case 2: \(-(T - 50) \leq 22\) simplifies to \(T \geq 28\)

This means Albany's average temperatures range from \[28, 72\] degrees Fahrenheit.
Tips for tackling inequalities:- **Write** each case clearly.- **Check** your solutions by substituting them back into the inequality.- **Always interpret** your results in the context of the problem.

Climate Comparison Using Inequalities

Inequalities can help reveal and compare climate characteristics of different regions by demonstrating the range of temperatures they experience.
Comparing San Francisco and Albany:

• **Wider Range**: Albany experiences a broader temperature range (\(28\) to \(72\) degrees) compared to San Francisco's tighter range (\(50\) to \(64\) degrees).
• **Colder Winters**: Albany's minimum average temperature is much lower (\(28\) degrees vs. San Francisco's \(50\) degrees), indicating colder winters.
• **Hotter Summers**: Albany also sees higher maximum averages (\(72\) degrees vs. \(64\) degrees for San Francisco), suggesting hotter summers.

These inequalities illustrate how math can provide insight into real-world differences in climate. Understanding these can be useful for decisions ranging from travel planning to studying climate change.