Problem 125 - Algebra for College Students

Question

Dr. Mosely has determined that \(99 \%\) of the babies he has delivered have weighed \(x\) pounds, where $$ |x-8.3|<1.5 $$ What range of weights corresponds to this inequality?

Step-by-Step Solution

Answer

The weights range from 6.8 pounds to 9.8 pounds.

Step 1: Understand the absolute value inequality

The given inequality is \(|x - 8.3| < 1.5\). This can be interpreted as the distance between \(x\) and 8.3 being less than 1.5.

Step 2: Convert to a compound inequality

Rewrite the absolute value inequality \(|x - 8.3| < 1.5\) as a compound inequality: \-1.5 < x - 8.3 < 1.5\.

Step 3: Solve the compound inequality

Add 8.3 to all parts of the compound inequality to solve for \(x\): \-1.5 + 8.3 < x < 1.5 + 8.3\. Simplifying this gives: \6.8 < x < 9.8\.

Step 4: Write the range of weights

The range of weights that satisfies the inequality \(|x - 8.3| < 1.5\) is between 6.8 pounds and 9.8 pounds.

Key Concepts

Understanding Compound Inequality

In the given problem, we start with an absolute value inequality, \(|x - 8.3| < 1.5\). This means the distance between \(x\) and 8.3 is less than 1.5. To solve this, we convert it into a compound inequality. This involves expressing it in a form without the absolute value, thus creating two separate inequalities. In this case, \(|x - 8.3| < 1.5\) translates to \(-1.5 < x - 8.3 < 1.5\). Now, this is a compound inequality because it has two parts that need to be satisfied simultaneously. Think of it as two separate conditions that \(x\) must meet.

Solving Inequalities

To solve the compound inequality, we deal with each part separately to isolate \(x\). Start by adding 8.3 to all parts of the inequality to balance the equation. So, \(-1.5 < x - 8.3 < 1.5\) becomes \(-1.5 + 8.3 < x < 1.5 + 8.3\).
Simplify the expressions on both sides: \6.8 < x < 9.8\.
This step-by-step adds clarity. By following simple arithmetic rules, like adding the same number to each part of the inequality, you properly isolate and find the value that \(x\) can take.

Finding the Range of Values

The final answer \6.8 < x < 9.8\ represents the range of values \(x\) can take, meaning the weights of the babies delivered. This range tells us that the weights fall between 6.8 pounds and 9.8 pounds.
Understanding the range of values helps in identifying the set of possible solutions within a specified limit. In this context, it means 99% of the babies weighed could be anywhere from 6.8 to 9.8 pounds. Imagine this range as a road with endpoints 6.8 and 9.8 — every point or weight on this road satisfies the original condition.