Problem 23 - Prealgebra

Question

Write and solve an inequality. You hike along the Appalachian Trail at 3 miles per hour. You stop for one hour for lunch. You want to walk at least 18 miles. How many hours should you expect to spend on the trail?

Step-by-Step Solution

Answer

You should spend at least 7 hours on the trail.

Step 1: Define Variables

Let the total time spent hiking be represented by \( t \) hours. The actual time spent hiking, excluding the lunch hour, is \( t - 1 \) hours.

Step 2: Write the Inequality

Since hiking is done at a speed of 3 miles per hour, in \( t - 1 \) hours you would cover \( 3(t - 1) \) miles. According to the problem, you want to cover at least 18 miles. Therefore, the inequality representing this situation is: \[ 3(t - 1) \geq 18 \]

Step 3: Simplify the Inequality

To solve the inequality, first distribute the 3: \[ 3t - 3 \geq 18 \]

Step 4: Isolate the Variable Term

Add 3 to both sides to isolate the variable term: \[ 3t \geq 21 \]

Step 5: Solve for t

Divide both sides by 3 to solve for \( t \): \[ t \geq 7 \]

Step 6: Conclusion

You should expect to spend at least 7 hours on the trail to walk at least 18 miles, including the 1-hour stop for lunch.

Key Concepts

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating these symbols. In our inequality problem, algebra is used to formulate a mathematical statement that describes the situation. Here, we express how a distance traveled relates to the time hiked. By utilizing an inequality, we can represent situations where a range of values is possible, rather than just a single value.
Understanding how to construct and manipulate algebraic expressions is crucial. It allows us to:

• Define relationships between numbers and expressions.
• Solve for unknown variables.
• Set up and solve equations and inequalities based on real-world scenarios.

Algebra allows students to approach problem-solving in an organized and effective way. Once you understand the method, manipulating equations and inequalities becomes a systematic process that can be applied to countless scenarios.

Problem-Solving

Problem-solving is a critical skill in mathematics. It involves understanding the problem, defining variables, setting up equations or inequalities, and then finding a solution. In this exercise, our goal was to determine how much time we need to spend hiking in order to cover at least 18 miles.
Here are the steps we followed:

• Defined the problem: We must hike at least 18 miles.
• Represented this scenario using mathematical expressions and inequalities.
• Simplified the inequality and isolated the variable by manipulating the expression using algebraic rules.
• Arrived at the solution, which tells us the minimum time needed to achieve our goal.

Good problem-solving employs logical reasoning. It shifts from chaotic guesswork to a straightforward methodology, ensuring solutions are both accurate and efficient.

Variables

Variables are symbols used to represent unknown values or quantities that can change. They are fundamental to algebra and essential for writing expressions and equations. In the context of our inequality problem, the variable \( t \) is used to represent the total time spent hiking.
Why do we use variables?

• They allow us to generalize problems and solutions.
• They make complex problems simpler to handle by representing unknown quantities as symbols.
• They help translate real-world problems into mathematical equations that can be solved systematically.

To effectively use variables:- Clearly define what each variable represents at the start of solving the problem.- Manipulate variables using known algebraic operations to isolate and solve for the variable.Employing variables in problem-solving enables a more dynamic approach, as they represent a broad range of scenarios rather than fixed numbers. This makes algebra a flexible and powerful tool in mathematics.