Problem 2 - Calculus for Biology and Medicine

Question

Find the values of \(x\) such that $$ |3 x-9|<0.01 $$

Step-by-Step Solution

Answer

The values of \(x\) that satisfy the inequality are in the interval \((2.9967, 3.0033)\).

Step 1: Understanding Absolute Value Inequality

To solve the inequality \(|3x - 9| < 0.01\), we start by recalling that an absolute value inequality of the form \(|u| < a\) means \(-a < u < a\). In this problem, \(u = 3x - 9\) and \(a = 0.01\).

Step 2: Write the Compound Inequality

From the inequality \(|3x - 9| < 0.01\), we write the compound inequality:\[ -0.01 < 3x - 9 < 0.01 \]

Step 3: Solve the Left Part of the Inequality

Solve \(-0.01 < 3x - 9\):1. Add 9 to both sides to isolate the term involving \(x\): \[ -0.01 + 9 < 3x \] \[ 8.99 < 3x \]2. Divide both sides by 3: \[ x > rac{8.99}{3} \] \[ x > 2.9967 \]

Step 4: Solve the Right Part of the Inequality

Solve \(3x - 9 < 0.01\):1. Add 9 to both sides: \[ 3x < 0.01 + 9 \] \[ 3x < 9.01 \]2. Divide both sides by 3: \[ x < rac{9.01}{3} \] \[ x < 3.0033 \]

Step 5: Combine the Solutions

Combine the results from Step 3 and Step 4 to find the interval for \(x\):\[ 2.9967 < x < 3.0033 \]

Key Concepts

Compound Inequalities

Compound inequalities involve two separate inequalities united by the word "and" or "or." In math, these compound statements reveal where a variable lies within specified limits. For our inequality, the absolute value form \(|3x - 9| < 0.01\)translates into the compound form \(-0.01 < 3x - 9 < 0.01\).This hinges on the property that \(|u| < a\) implies \(-a < u < a\).

Here, it's all about finding a range of values for \(x\).

• The number \(3x - 9\) falls between two bounds: -0.01 and 0.01.
  • This tells us precisely how close \(3x - 9\) needs to be to 0.
• Instead of a single outcome, you achieve an interval.

When we set up a compound inequality like this, we solve for both edges, thus pinning the variable \(x\) neatly between two definitive values.Understanding this concept is crucial in properly manipulating absolute values into solvable inequalities.

Solving Inequalities

Solving inequalities involves determining the variable's possible values that satisfy the equation's rules. The objective is, essentially, to isolate \(x\). Starting with our compound inequality \(-0.01 < 3x - 9 < 0.01\), we break it into two solid parts:

• The left part: \(-0.01 < 3x - 9\)
• The right part: \(3x - 9 < 0.01\)

The trick lies in handling each inequality segment individually:

• Add 9 to both sides, transferring numbers so that \(x\) is left standing alone.
• Divide by 3 to completely solve for \(x\) in each section.

These steps maintain the equation's integrity, yet offer clarity on how the inequality transitions from absolute value into a simple interval for \(x\). You follow the process for each section and then unite them to display the final solution. This yields understanding not just of the equation's solution but of the values \(x\) plays within.

Algebraic Techniques

Algebraic techniques are essential for untangling intricate problems like inequalities. When we talk about solving the inequality \(|3x - 9| < 0.01\), several algebra skills come into play.First, the absolute value handling shifts into a compound inequality using the basic principle that makes \(-0.01 < 3x - 9 < 0.01\) possible from \(|3x - 9| < 0.01\).

You utilize fundamental skills like:

• Adding or subtracting numbers to keep the equation balanced while pushing coefficients to one side.
• Dividing through by constants, a crucial step for isolating \(x\).

Each of these actions transforms the initial **absolute value inequality** into manageable parts. The algebraic manipulation allows transitions from broad expressions to sharp intervals. Building a disciplined approach through algebra makes solutions not only possible, but clean, clear, and logical.Understanding and applying these algebraic techniques is not just about solving the problem, but mastering the basics that run through every algebra study.