Problem 4 - Calculus for Biology and Medicine

Question

Find the values of \(x\) such that $$ |2 \sqrt{x}-5|<0.01 $$

Step-by-Step Solution

Answer

The solution is \(6.225025 < x < 6.275025\).

Step 1: Understand the absolute inequality

The given inequality is \(|2 \sqrt{x} - 5| < 0.01\). This means the expression \(2 \sqrt{x} - 5\) must lie between \(-0.01\) and \(0.01\).

Step 2: Break down the inequality

The absolute value inequality \(|2 \sqrt{x} - 5| < 0.01\) can be split into two separate inequalities:1. \(2 \sqrt{x} - 5 < 0.01\)2. \(2 \sqrt{x} - 5 > -0.01\)

Step 3: Solve the first inequality

Solve \(2 \sqrt{x} - 5 < 0.01\) by isolating \(2 \sqrt{x}\):\[ 2 \sqrt{x} < 5 + 0.01 \]\[ 2 \sqrt{x} < 5.01 \]Now divide both sides by 2:\[ \sqrt{x} < 2.505 \]Square both sides:\[ x < 2.505^2 \]\[ x < 6.275025 \]

Step 4: Solve the second inequality

Solve \(2 \sqrt{x} - 5 > -0.01\) by isolating \(2 \sqrt{x}\):\[ 2 \sqrt{x} > 5 - 0.01 \]\[ 2 \sqrt{x} > 4.99 \]Now divide both sides by 2:\[ \sqrt{x} > 2.495 \]Square both sides:\[ x > 2.495^2 \]\[ x > 6.225025 \]

Step 5: Combine the solutions

The solution to the original inequality is the intersection of the two intervals found:\(x > 6.225025\) and \(x < 6.275025\).Thus, the values of \(x\) such that \(|2 \sqrt{x} - 5| < 0.01\) are:\[ 6.225025 < x < 6.275025 \]

Key Concepts

Solving Inequalities

Inequalities are mathematical expressions involving the symbols ">", "<", "≥", and "≤". Solving inequalities involves finding the range of values that make the inequality true. For example, in the inequality \(|2 \sqrt{x} - 5| < 0.01\), we want to find all the values of \(x\) that satisfy this condition. To do this, we must break the absolute value inequality into two separate inequalities. This is because absolute value expressions measure the distance of a number from zero without regard to direction, meaning we need to consider both the positive and negative possibilities. When we break down \(|2 \sqrt{x} - 5| < 0.01\), it gives us two straightforward inequalities:

• \(2 \sqrt{x} - 5 < 0.01\)
• \(2 \sqrt{x} - 5 > -0.01\)

This lets us focus separately on the conditions when the expression is slightly above and slightly below the given range. Solving each separately before combining the solutions gives the final answer.

Square Roots

Working with square roots is critical in solving inequalities involving radicals. A square root essentially asks what number, when multiplied by itself, results in the original number under the radical. For instance, in our scenario, we are dealing with \(\sqrt{x}\) when solving the inequalities. To isolate \(\sqrt{x}\) in \(2 \sqrt{x} - 5 < 0.01\), we first adjust the inequality so that the radical part is alone on one side. For example:

• After isolating, we find \(2 \sqrt{x} < 5.01\).
• Then divide by 2 to get \(\sqrt{x} < 2.505\).

At this point, we can clear the square root by squaring both sides of the inequality, resulting in the simplified form \(x < 6.275025\). Similarly, for the other inequality, \(2 \sqrt{x} - 5 > -0.01\), we perform similar steps:

• Manipulate the inequality to get \(\sqrt{x} > 2.495\).
• Square both sides to find \(x > 6.225025\).

This process demonstrates the efficient use of square roots to solve parts of an inequality.

Interval Notation

Interval notation is a concise way to describe a set of numbers between given boundaries. When we arrive at the solution of an inequality, such as \(6.225025 < x < 6.275025\), it can be neatly expressed using interval notation. In the solution to our original inequality, the possible values of \(x\) range between these two boundaries, excluding the endpoints. This type of range is an *open interval*, described as \((6.225025, 6.275025)\) in interval notation. Understanding interval notation is crucial for expressing solutions efficiently:

• Parentheses \(()\) indicate that the endpoint values are not included (open interval).
• Brackets \([]\) indicate the endpoints are included (closed interval).

A mastery of interval notation allows for precise and clear communication of solution sets in mathematics.