Problem 149 - College Algebra

Question

Determine whether 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

Yes, the statement makes sense. The given expression \(|x-20| \leq 5\) correctly models the condition for winning the contest as it restricts the completion time, \(x\), to a range between 15 and 25 minutes, which is 20 minutes plus or minus 5 minutes.

Step 1: Exploring the Equation

Let's first understand what the equation \(|x-20| \leq 5\) means. This is an absolute value equation, where \(x\) is the variable. The absolute value of a number (or a difference) represents its distance from zero. Therefore, \(|x-20|\) represents the distance from \(x\) to 20. When this distance is less than or equal to 5, \(x\) will lie in the range 15 to 25.

Step 2: Matching the Equation with the Statement

Now we have to consider whether this equation matches the given scenario. The statement claims that to win a contest, a crossword puzzle must be completed within a time range of 15-25 minutes. This matches precisely with the equation, because \(x\), the completion time, lies between 15 and 25, i.e., 20±5.

Step 3: Testing an Exception

Let's just check a condition outside the defined range to see if it makes sense. For example, does it make sense if \(x=30\) (i.e., the crossword is completed in 30 minutes)? Putting \(x=30\) in the equation, we get \(|30-20| \leq 5\), or \(10 \leq 5\), which is not true. This means the model won't hold for values of \(x\) outside the 15-25 range, which aligns with the statement.

Key Concepts

Crossword Puzzle Time Limit

When tackling a crossword puzzle timed contest, it's important to account for variability in how fast you can finish. The statement provided suggests you want to complete the puzzle within a time frame of 20 minutes, give or take 5 minutes. This means your successful completion time, \(x\), should fall within the interval from 15 minutes to 25 minutes.
This range is derived using absolute value inequalities, which help us express conditions with flexibility, like time limits where you can be a bit faster or slower. It's as if you're stating, "I'm okay if I finish anywhere from 15 to 25 minutes," including both bounds. This approach is very useful in contests where a bit of variance is expected.
It's like setting a flexible deadline that still holds you to completing the task efficiently, ensuring you fit within the top competitor spectrum in that 10-minute window.

Solving Inequalities

Solving inequalities is an essential part of uncovering the logical limits or thresholds for a condition. In this exercise, the inequality \(|x-20| \leq 5\) helps model the desired completion time for the crossword puzzle.
In algebra, solving an inequality involves finding all the possible values of the variable that make the statement true.

• First, remove the absolute value by considering both possible "distances" from the point, leading to two scenarios: one where \(x-20\) is less than or equal to 5, and another where \(x-20\) is greater than or equal to -5.
• This results in two simpler inequalities: \(x \leq 25\) and \(x \geq 15\), which combine to describe a continuous range between 15 and 25 minutes.

This method allows us to visualize the problem's boundaries, ensuring that you comprehend both concrete and conceptual parts of such algebraic statements.

Distance Interpretation in Algebra

The concept of distance within algebra is often tied to the notion of absolute value. Abstactly, absolute value functions express the non-negative distance of a number from zero. When transposed to inequalities, it can interpret how far away variables are from a certain point or target.
In this exercise, \(|x-20|\) isn't just about numbers. It’s about timing—specifically, how far a potential completion time \(x\) can stray from the "ideal" 20 minutes while remaining a valid solution to the equation.
This idea serves many purposes:

• It simplifies complex real-world conditions into algebraic terms, giving us clear limits to work within.
• It embodies a real-life application where flexible ranges are necessary, much like estimating the maximum allowable deviation when calculating costs, speeds, or time frames.

By understanding this distance, you analyze not just the numeric relationship but its applied significance, a practice invaluable in mathematical problem-solving and beyond.