Problem 91 - Precalculus Student Solutions Manual 5th

Question

Solve each problem. Dr. Tydings has found that, over the years, \(95 \%\) of the babies he has delivered weighed \(x\) pounds, where $$|x-8.2| \leq 1.5.$$ What range of weights corresponds to this inequality?

Step-by-Step Solution

Answer

The range of weights is 6.7 to 9.7 pounds.

Step 1 - Understand the absolute value inequality

The inequality given is \[|x - 8.2| \leq 1.5.\]

Step 2 - Rewrite the absolute value inequality without absolute value

The absolute value inequality \[|x - 8.2| \leq 1.5\] can be rewritten as two separate inequalities: \[-1.5 \leq x - 8.2 \leq 1.5.\]

Step 3 - Solve the compound inequality

To solve the compound inequality \[-1.5 \leq x - 8.2 \leq 1.5,\] we first add 8.2 to all parts of the inequality:\[-1.5 + 8.2 \leq x \leq 1.5 + 8.2.\]

Step 4 - Simplify the inequality

Simplifying the left side: \[-1.5 + 8.2 = 6.7,\] and simplifying the right side: \[1.5 + 8.2 = 9.7.\] So, \[6.7 \leq x \leq 9.7.\]

Key Concepts

Compound Inequalities

When working with compound inequalities, you deal with two or more inequalities connected by 'and' or 'or'. In the exercise, we faced a compound inequality in the form of ewlineoften written as ewlineThese inequalities must be solved together.ewlineTo manage such compound inequalities, treat each part of the inequality separately. When you combine the solutions, you arrive at the range where the variable satisfies both inequalities simultaneously. This method is crucial when both parts affect the outcome; neglecting one can lead to incorrect results.

Solving Inequalities

Solving inequalities follows similar principles to solving regular algebraic equations, with a few additional rules.

• Isolate the variable on one side of the inequality
• Perform the same operations on both sides of the inequality
• If you multiply or divide by a negative number, reverse the inequality sign

In our exercise, the inequalities solved were:ewlineewlineto add or subtract the same amount to both sides of the inequalities to isolate the variable.ewlineFinally, verify your solutions by substituting them back into the original inequality to ensure they hold true.

Absolute Value Properties

Absolute value ewlineThe properties of absolute values are helpful in solving such inequalities. For instance:Property 1: ewlineProperty 2: ewlineewline ewlineStep 2 involves rewriting it as compound inequalities,ewlineThis understanding allows you to split absolute value inequalities into more accessible parts, leading to simpler algebraic manipulations.
Always remember to test your final solutions within the context of absolute value properties to confirm they meet the initial condition.Summarizing the exercise, the weights x pounds fall between 6.7 and 9.7 as derived ensures that all obtained solutions satisfy the condition .