Problem 1 - Calculus for Biology and Medicine

Question

Find the values of \(x\) such that $$ |2 x-1|<0.01 $$

Step-by-Step Solution

Answer

The solution is \(0.495 < x < 0.505\).

Step 1: Understand the Inequality

The inequality \(|2x-1|<0.01\) means that the expression inside the absolute value sign, \(2x-1\), must be less than 0.01 units away from zero on the number line. The absolute value inequality can be split into two separate inequalities.

Step 2: Split the Absolute Value Inequality

The inequality \(|2x-1|<0.01\) can be split into two separate inequalities without the absolute value sign: \(2x - 1 < 0.01\) and \(2x - 1 > -0.01\). Solving each of these inequalities separately will give the range of values for \(x\).

Step 3: Solve the Inequality \(2x - 1 < 0.01\)

To solve \(2x - 1 < 0.01\), first add 1 to both sides to get \(2x < 1.01\). Then divide both sides by 2 to isolate \(x\), which results in \(x < 0.505\).

Step 4: Solve the Inequality \(2x - 1 > -0.01\)

To solve \(2x - 1 > -0.01\), first add 1 to both sides to get \(2x > 0.99\). Then divide both sides by 2 to isolate \(x\), resulting in \(x > 0.495\).

Step 5: Combine the Solutions

The solutions from Steps 3 and 4 can be combined to give a range for \(x\). So, \(x\) must satisfy both \(x > 0.495\) and \(x < 0.505\). Therefore, the solution is \(0.495 < x < 0.505\).

Key Concepts

Inequality Solving

Inequality solving is a fundamental aspect of mathematics that helps us find a range of values for a variable satisfying a given condition. Unlike equations, which show equality, inequalities indicate that one expression is greater or less than another. They come with signs like greater than (\(>\)), less than (\(<\)), or their non-strict cousins, greater than or equal to (\(\geq\)) and less than or equal to (\(\leq\)).
To solve inequalities, consider them as questions asking "how far can the values stretch?". Just like balancing a scale, whatever is done to one side must be done to the other:

• Add or subtract the same number from both sides.
• Multiply or divide both sides by the same positive number without changing the inequality sign.
• If you multiply or divide by a negative number, flip the inequality sign.

Learning these rules helps to neatly find the sets of possible solutions, just like lining up dominoes ready to fall into correct order.

Absolute Value

Absolute value tells us how far away a number is from zero, without considering direction. Represented with vertical bars \(|x|\), the absolute value of a number is its distance from zero on the number line. For any given number, positive or negative, its absolute value is always non-negative, because distance can't be negative.
When solving inequalities involving absolute values, the goal is to determine the possible numbers inside these bars that fit the condition. In essence, absolute value inequalities, like \(|a| < b\), can be split into two scenarios to manage this distance from zero:

• \(a < b\)
• \(a > -b\)

These two conditions combined will guide us in a gentle direction towards the correct range of values, leading you to find solutions that satisfy all parts of the inequality.

Mathematics Problem Solving

Mathematics problem solving is akin to finding your way through a maze. It involves understanding the problem, strategizing a plan, carrying out that plan, and reviewing the results. Let's consider each step which is crucial in unraveling math problems:

• Understanding the Problem: Identify what is being asked. In our case, it's recognizing we're dealing with absolute value inequalities.
• Strategizing: Break down the overarching issue. For absolute value, this means splitting it into two simpler inequalities.
• Execution: Work step-by-step. Solve each inequality separately to isolate the variable.
• Review: Ensure your solution meets all specified conditions and makes logical sense. Combine the intervals found from each individual inequality to form a solution set.

This systematic approach not only helps in solving equations but cultivates a structured mind, enabling you to tackle more complex problems in mathematics and beyond.