Question

Physicians consider an adult's body temperature \(x\) (in degrees Fahrenheit) to be normal if it satisfies the inequality \(|x-98.6| \leq 1\) Determine the range of temperatures that are considered to be normal.

Short answer

The range of normal body temperatures for adults is between 97.6 degrees Fahrenheit and 99.6 degrees Fahrenheit, inclusive.

Step-by-step solution

Understand Absolute Value Inequality

An absolute value inequality like \(|x-98.6| \leq 1\) describes two separate scenarios due to the nature of absolute values: one where \(x-98.6\) is less than or equal to 1, and one where \(x-98.6\) is greater than or equal to -1. Therefore, we can rewrite the inequality as two separate inequalities: \(x - 98.6 \leq 1\) and \(x - 98.6 \geq -1\).

Solve each inequality

To solve the inequalities, isolate \(x\) in each equation. For \(x - 98.6 \leq 1\), add 98.6 to both sides to get \(x \leq 99.6\). For \(x - 98.6 \geq -1\), add 98.6 to both sides to get \(x \geq 97.6\).

Define the range

These two inequalities indicate that the normal range of body temperatures \(x\) (in degree Fahrenheit) is all numbers between 97.6 and 99.6 inclusive. We write this as \(97.6 \leq x \leq 99.6\).

Key concepts

Solving Inequalities

Understanding how to solve inequalities is a foundational algebra skill that enables students to analyze ranges of values within real-world contexts. Inequalities let us express conditions where there's more than one acceptable answer, for example, the range of normal body temperatures.

Let's take the given absolute value inequality, \(|x - 98.6| \leq 1\), which represents all the temperatures that are within 1 degree of the normal 98.6 degrees Fahrenheit. To solve it, we looked at the two scenarios the absolute value could represent: \(x - 98.6 \leq 1\) and \(x - 98.6 \geq -1\).

Solving these two inequalities gave us the range \(97.6 \leq x \leq 99.6\), meaning any body temperature between 97.6°F and 99.6°F is considered normal. Understanding inequalities can thus be used to solve practical problems, like finding acceptable human body temperatures.

Body Temperature Range

When we talk about body temperature range in a medical context, we refer to a spectrum of temperatures that are recognized as healthy for the human body. Typically, the normal core body temperature is around 98.6°F (37°C), but this can fluctuate during the day and vary from person to person.

In our exercise, the body temperature range is described by the inequality \(|x - 98.6| \leq 1\), which translates to a range of 97.6°F to 99.6°F being considered normal. This range is crucial for physicians as it helps in diagnosing potential health issues. A temperature outside of this range might indicate a fever or hypothermia. It's the algebraic understanding of inequalities that allows us to accurately represent and calculate these ranges.

Algebraic Reasoning

Algebraic reasoning is the process of identifying patterns, making conjectures, and using symbols and formulas to solve problems. It is a thought process that helps in developing a deeper understanding of mathematical concepts such as inequalities.

In this context, algebraic reasoning was used to dissect the meaning behind the absolute value inequality. This required breaking down the problem into two scenarios and solving for the variable \(x\) in each case. The reasoning process was crucial to interpret the absolute value as a measure of 'distance' from 98.6°F and determine the range of acceptable temperatures. Good algebraic reasoning leads to accurate conclusions and is central to dealing with real-world mathematical scenarios, such as determining a normal body temperature range.