Problem 128 - Algebra for College Students

Question

The average clotting time of blood is \(7.45 \mathrm{sec}\), with a variation of plus or minus \(3.6 \mathrm{sec}\). Write this statement as an absolute value inequality, with \(x\) representing the time. Solve the inequality.

Step-by-Step Solution

Answer

The absolute value inequality based on the given information is: This means that the clotting time should be between 3.85 seconds and 11.05 seconds to satisfy the condition.

Step 1: Identify the given quantities

The average clotting time of blood is given as 7.45 seconds, and the variation is given as plus or minus 3.6 seconds. We’ll use these quantities to set up the absolute value inequality.

Step 2: Formulate the expression

Given that the clotting time, represented by the variable x, should not deviate more than 3.6 seconds from the average of 7.45 seconds, this can be written as:

Step 3: Set up the absolute value inequality

This scenario can be written as an absolute value inequality:

Key Concepts

average clotting time

The average clotting time of blood signifies the typical duration required for blood to clot in an ideal scenario. In this particular exercise, the average clotting time is given to be 7.45 seconds. This means that, on average, healthy individuals have their blood clot within approximately this time frame. Understanding the average clotting time helps medical professionals monitor and diagnose conditions related to blood clotting.
Considering the word 'average,' it means that individual clotting times can vary around this central value due to natural fluctuations in biological processes.

variation in time

Variation in time refers to how much individual clotting times can differ from the average clotting duration. In this exercise, the variation around the average clotting time of 7.45 seconds is given as plus or minus 3.6 seconds.
This indicates that some blood clotting times could be as low as 3.85 seconds (7.45 - 3.6) or as high as 11.05 seconds (7.45 + 3.6). Variation is essential in statistics and helps in understanding the spread of data points around the mean value.
In medical testing, acknowledging this variation is crucial, as it ensures that the data reflects real-world scenarios and accounts for natural differences among individuals.

solving inequalities

Solving inequalities involves finding the range of values that satisfy a given condition. In this exercise, we aim to express the clotting time variation as an absolute value inequality and then find the possible range for the clotting time (x).
Let's first understand the absolute value inequality:

• Absolute value notation represents the distance of a number from zero on the number line.
• For example, if we say \(|x| < 3.6\), it means x can be any number within 3.6 units of 0.

Now, we can write the clotting time scenario as: \(| x - 7.45 | \leq 3.6\). This inequality states that the difference between the clotting time (x) and the average clotting time (7.45 seconds) should not exceed 3.6 seconds.
To solve this, follow these steps: Break down the absolute value inequality into two separate inequalities: \( x - 7.45 \leq 3.6 \) and \(- (x - 7.45) \leq 3.6 \) [which simplifies to \( x - 7.45 \geq -3.6 \)]. Then, solve each part for x to find the range of acceptable values for the clotting time.
This results in: \( 3.85 \leq x \leq 11.05 \). Therefore, clotting times can vary from 3.85 seconds to 11.05 seconds while staying within the allowed variation from the average clotting time.