Problem 3 - Calculus for Biology and Medicine

Question

Find the values of \(x\) such that $$ \left|x^{2}-9\right|<0.1 $$

Step-by-Step Solution

Answer

\(-3.016 < x < -2.983\) or \(2.983 < x < 3.016\).

Step 1: Understand Absolute Inequality

The given inequality is \( |x^2 - 9| < 0.1 \). This tells us that the expression \( x^2 - 9 \) must be less than 0.1 units away from zero. This can be represented as a double inequality: \( -0.1 < x^2 - 9 < 0.1 \).

Step 2: Solve the Lower Bound

Start by solving the lower bound of the inequality: \( -0.1 < x^2 - 9 \). This rearranges to \( x^2 > 8.9 \).

Step 3: Solve the Upper Bound

Next, solve the upper bound: \( x^2 - 9 < 0.1 \). This rearranges to \( x^2 < 9.1 \).

Step 4: Combine the Inequalities

Now we combine the two inequalities obtained from Steps 2 and 3: \( 8.9 < x^2 < 9.1 \). This means \( x^2 \) is between 8.9 and 9.1.

Step 5: Solve for x

To find \(x\), take the square root of each part of the inequality: \( \ \sqrt{8.9} < |x| < \ \sqrt{9.1} \). This further splits into two inequalities: \( \ -\sqrt{9.1} < x < -\sqrt{8.9} \) and \( \ \sqrt{8.9} < x < \sqrt{9.1} \).

Step 6: Calculate Numerical Values

Estimate the square roots: \( \sqrt{8.9} \approx 2.983 \) and \( \sqrt{9.1} \approx 3.016 \). Thus, the solution is \( -3.016 < x < -2.983 \) or \( 2.983 < x < 3.016 \).

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions used to show the relationship of inequality between two values or expressions. In an inequality, instead of an equal sign, we use signs like `>` (greater than), `<` (less than), `≥` (greater than or equal to), and `≤` (less than or equal to). These symbols help us understand how the values compare to each other.

When dealing with inequalities, it's essential to recognize some key steps:

• Identify the type of inequality and the expression involved.
• Understand how the inequality splits into different regions on the number line.
• Solve the inequality by manipulating it similarly to an equation, but with different rules for when you multiply or divide by negative numbers.

In the given exercise, the inequality is absolute: \( |x^2 - 9| < 0.1 \). This specifies that the value of \( x^2 - 9 \) is confined within a specific range from zero, allowing us to write it as a double inequality \( -0.1 < x^2 - 9 < 0.1 \), thus restricting \( x^2 \) to a very narrow interval.

Square Roots in Inequalities

Square roots are a crucial concept when solving inequalities involving squared terms like \( x^2 \). The square root function helps find a value that, when squared, gives the original number. Notably, square roots are always non-negative.

When dealing with an inequality such as the one in the problem, where you need to determine \( x \) from bounds expressed in terms of \( x^2 \), the square root is your tool of choice:

• Take the square root of each side of the inequality separately.
• Remember that squaring and square rooting are inverse operations.
• Incorporate both the positive and negative solutions since both contribute to the solution interval.

For the inequality \( 8.9 < x^2 < 9.1 \), we apply the square root to obtain intervals for \( x \): \( \sqrt{8.9} < |x| < \sqrt{9.1} \). This results in two scenarios: positive and negative intervals for \( x \).

Using Mathematical Reasoning

Mathematical reasoning is the process of logical thinking to deduce truths from premises and arrive at conclusions based on a set of facts or conditions. It's like detective work for numbers. To solve any problem effectively, especially involving inequalities, one must rely on clear reasoning steps.

Here's the important part of applying mathematical reasoning:

• Start by understanding what the problem is asking and the constraints it imposes.
• Break down the problem into manageable parts, as shown in the steps: recognizing bounds, transforming absolute value constraints into double inequalities.
• Consider alternate solutions by exploring positive or negative roots, as square equations often yield multiple solutions.
• Verify the results to ensure they make sense within the original problem's context.

In this exercise, you are systematically guided through transforming an absolute inequality into solvable conditions that ultimately reveal the constrained range of \( x \). This not only confirms understanding of each transformation but also illustrates the effectiveness of systematic reasoning.