Problem 113 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. Dr. Cazayoux has found that, over the years, \(95 \%\) of the babies he delivered weighed \(x\) pounds, where \(|x-8.0| \leq 1.5 .\) What range of weights corresponds to this inequality?

Step-by-Step Solution

Answer

The weight range is from 6.5 to 9.5 pounds.

Step 1: Understanding the Inequality

The given inequality is \(|x - 8.0| \leq 1.5\). This expression represents the absolute difference between the weight of babies, \(x\), and 8.0 pounds, which should be less than or equal to 1.5 pounds.

Step 2: Breaking Down the Absolute Value Equation

The inequality \(|x - 8.0| \leq 1.5\) can be rewritten as two separate inequalities: 1. \(x - 8.0 \leq 1.5\) 2. \(x - 8.0 \geq -1.5\) This breaks the absolute value condition into manageable parts.

Step 3: Solving the First Inequality

Solve the inequality \(x - 8.0 \leq 1.5\) by adding 8.0 to both sides:\[x \leq 1.5 + 8.0 \implies x \leq 9.5\]

Step 4: Solving the Second Inequality

Solve the inequality \(x - 8.0 \geq -1.5\) by adding 8.0 to both sides:\[x \geq -1.5 + 8.0 \implies x \geq 6.5\]

Step 5: Compiling the Range of Weights

Combine the results from the two inequalities to find the range of values for \(x\): \[6.5 \leq x \leq 9.5\]This means the weights of the babies ranged from 6.5 pounds to 9.5 pounds.

Key Concepts

Understanding Absolute Value

Absolute value is a fundamental concept in mathematics that helps capture the idea of distance in a straightforward manner. When we talk about the absolute value of a number, we are discussing its distance from zero on a number line. This is always a positive quantity or zero.
Absolute value is denoted using vertical bars, like this: \(|x|\).
For example, \(|-4| = 4\) because -4 is four units away from zero, and similarly, \(|4| = 4\) due to being equidistant in the opposite direction.

• The absolute value of a positive number is the number itself.
• The absolute value of zero is zero.
• The absolute value of a negative number is its positive counterpart.

In the context of the exercise \(|x - 8.0| \leq 1.5\), the expression signifies that the difference between the baby's weight and 8.0 pounds must be within a \(1.5\) pound range, either above or below.
This can be thought of as creating a safe zone around a specific value—in this case, 8.0 pounds, where Dr. Cazayoux's delivered babies' weights mostly fall.

Applying Mathematical Modeling

Mathematical modeling is an essential tool in translating real-world problems into mathematical language. It involves representing scenarios with numbers, symbols, and equations to analyze the situation and draw conclusions.
In the problem presented, Dr. Cazayoux uses modeling to statistically describe the weight characteristics of newborns he has delivered over time.
Let's break down the process:

• The known value, 8.0 pounds, is modeled as a reference or typical weight for the babies.
• The inequality \(|x - 8.0| \leq 1.5\) captures the range of variability around this typical weight. The inequality symbolizes weights fluctuating within a narrow band from 6.5 to 9.5 pounds.

This model is built on using absolute value to express the concept of deviation from a norm. By solving it, we obtain practical insights into real-life phenomena such as birth weights.

Problem Solving Techniques

Solving absolute value inequalities can seem challenging at first, but it becomes manageable when broken into simpler components.
Let's discuss how the solution was derived for \(|x - 8.0| \leq 1.5\):

1. Recognize that the absolute value inequality resolves into two separate statements: \(x - 8.0 \leq 1.5\) and \(x - 8.0 \geq -1.5\).
2. For \(x - 8.0 \leq 1.5\): solving by isolating \(x\) means adding \(8.0\) to both sides, giving us \(x \leq 9.5\).
3. For the second statement, \(x - 8.0 \geq -1.5\): similarly solve by adding \(8.0\), to get \(x \geq 6.5\).
4. Finally, combine both inequalities to define the solution set, \(6.5 \leq x \leq 9.5\). This combined inequality readily gives the range of weights the doctor expects.

Good problem-solving skills involve breaking down complex tasks, understanding the needed mathematical operations, and reassembling those parts to form a coherent solution. This systematic approach enhances comprehension and ensures accuracy in problem-solving.