Problem 48 - Calculus with Concepts in Calculus

Question

Solve the equation. $$ |x|=\pi $$

Step-by-Step Solution

Answer

The solutions are \(x = \pi\) and \(x = -\pi\).

Step 1: Understanding Absolute Value

The absolute value of a number is always positive. It is the distance of a number from zero on the number line. Therefore, if \(|x| = \pi\), it means the distance from zero is \(\pi\).

Step 2: Setting Up Two Equations

Since the absolute value represents the distance, \(|x| = \pi\) implies there are two possible values for \(x\): \(x = \pi\) and \(x = -\pi\). These are the two equations we need to solve.

Step 3: Solving First Equation

The first equation is \(x = \pi\). Since \(\pi\) is an irrational number, this solution is simply \(x = \pi\).

Step 4: Solving Second Equation

The second equation is \(x = -\pi\). This solution is also straightforward: \(x = -\pi\).

Key Concepts

Irrational Numbers

Irrational numbers are interesting digital beings! They are numbers that cannot be expressed as a simple fraction. They go on and on forever without repeating. One of the most famous irrational numbers is \( \pi \), which stands for the ratio of the circumference of a circle to its diameter. It starts out as 3.14159 and continues without an end.

• Other examples of irrational numbers include \( \sqrt{2} \) and \( e \) (the base of the natural logarithm).
• Since you can't neatly write irrational numbers as a fraction, they are known as having non-repeating, non-terminating decimal forms.

In the solution above, \( \pi \) is an irrational number, and it shows up in this absolute value equation. This is the reason why direct numerical computation for the exact value isn’t possible. Instead, we rely on understanding its approximations and properties.

Distance on Number Line

The number line is a great way to visualize numbers. It stretches indefinitely in both directions, with each point representing a number. The absolute value of any number is its distance from zero on this line, without worrying whether the number is to the right or left of zero.

In the equation given above, \(|x| = \pi\), means the distance of \(x\) from zero is \(\pi\). Think about it as steps away from zero:

• \( x = \pi \) would be \(\pi\) steps to the right.
• \( x = -\pi \) would be \(\pi\) steps to the left.

These two numbers are equidistant from zero, showing why absolute value gives us two solutions. By understanding the distance, we get a picture of where \( x \) could be on the number line. This concept helps explain why we get both positive and negative results when dealing with absolute value equations.

Absolute Value Properties

At the heart of this problem is the concept of absolute value. Absolute value tells you how far a number is from zero, without considering direction. Here are some key properties you should know:

• \( |x| = x \) if \( x \geq 0 \), meaning the number is positive or zero.
• \( |x| = -x \) if \( x < 0 \), meaning the number is negative, but the result is positive due to the property of absolute values.
• Absolute values are always non-negative.
• It allows us to consider both positive and negative solutions, as shown by the equations \( x = \pi \) and \( x = -\pi \).

Understanding these properties is crucial when solving absolute value equations. They help us set up correct equations to find all possible solutions. In our exercise, we utilized these properties to determine the two potential outcomes for \( x \). This concept ensures that we don't miss a solution on the number line.