Problem 149 - Algebra and Trigonometry

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'll win the contest if I can complete the crossword puzzle in 20 minutes plus or minus 5 minutes, so my winning time, \(x\), is modeled by \(|x-20| \leq 5\)

Step-by-Step Solution

Answer

The statement makes sense because the mathematically modeled time range (15 to 25 minutes) perfectly aligns with the described scenario in which a participant can win by completing the crossword puzzle within 20 minutes, plus or minus 5 minutes.

Step 1: Understanding the inequality

An absolute value inequality \(|x - a| \leq b\) includes all values of \(x\) that are at most \(b\) units away from \(a\). In this case, \(|x - 20| \leq 5\), means \(x\) is at most 5 minutes away from 20 minutes.

Step 2: Unpacking the inequality

This inequality can be rewritten as a compound inequality: \(15 \leq x \leq 25\). This statement includes all numbers \(x\) that are between 15 and 25 inclusive.

Step 3: Comparing the inequality to the problem statement

The problem statement says that to win the contest, the crossword puzzle must be completed within 20 minutes plus or minus 5 minutes. This time range is 15 minutes to 25 minutes, inclusive. This is the exact interval represented by the inequality in Step 2.

Step 4: Final assessment

Since the time range presented in the problem and the time range represented by inequality match, we can conclude that the statement makes sense.

Key Concepts

Compound Inequality

A compound inequality is essentially two inequalities joined together by either an "and" or an "or". When you come across an absolute value inequality like \( |x - 20| \leq 5 \), it can be split into a compound inequality. Here, the absolute value inequality \( |x - 20| \leq 5 \) translates into \( 15 \leq x \leq 25 \. \) This is because the inequality expresses that \( x \) is within 5 units from 20 on the number line.

• The endpoint values (15 and 25 here) are included in the range.
• This forms a closed interval, represented mathematically with brackets.

Understanding compound inequalities allows us to see how absolute values set a range within which a variable can exist.
Think of this as creating a boundary or window, like setting limits within which an event or number is valid.

Interval Notation

Interval notation is a way of writing sets of numbers that represent a range. It is especially useful in algebra to clearly convey solutions for inequalities. In the exercise provided, our compound inequality \( 15 \leq x \leq 25 \) can be expressed in interval notation as \([15, 25]\).

• The square brackets \([ \text{and } ]\) mean that the end numbers are included in the set.
• If an end number was not included, we would use parentheses \(( \text{or } )\).

Interval notation provides a concise way to show all solutions that satisfy the inequality.
This makes it easier to visualize and understand the range of numbers we are working with in problems like this.
It streamlines communication and ensures clarity in mathematical discussions.

Problem Solving in Algebra

Solving algebraic problems requires breaking down the question, identifying relevant concepts, and applying appropriate mathematical principles.
In the exercise, this meant translating the word problem into a mathematical inequality and applying the concepts of absolute value and compound inequality.

• The problem provided a target time (20 minutes) and a tolerance allowance (+/- 5 minutes).
• By interpreting this range, we could establish a valid mathematical model through the inequality \( |x - 20| \leq 5 \).

The critical aspect of problem solving is understanding what the problem asks, deciding the right approach, and correctly executing the solution.
This often includes checking back to ensure calculations and interpretations make sense with the original context.
Practicing these steps will improve your algebraic problem-solving skills.

Mathematical Reasoning

Mathematical reasoning is the logical thought process that allows us to make sense of mathematical statements and solve problems effectively.
This involves understanding mathematical concepts, connecting them to the problem at hand, and reasoning through outcomes. Using the exercise:

• We recognized that the given time range influenced the solution method.
• The goal was to ensure the inequality represented this range accurately.

In determining whether the statement "I'll win the contest if I meet the specified time range" makes sense, we engaged a reasoning process:

• Translating the word problem into a understandable inequality.
• Re-evaluating the solution against the problem's conditions.
• Seeing the congruence between the mathematical solution and the problem context.

Effective mathematical reasoning is crucial in ensuring solutions don't just work mathematically but align with the real-world context of a problem.
This skill helps bridge the gap between theoretical math and practical application.