Problem 38 - Calculus: One and Several Variables

Question

Each of the following sets is the solution of an inequality of the form \(|x-c|<\delta\). Find \(c\) and \(\delta\). $$(-2,2)$$.

Step-by-Step Solution

Answer

The center of the interval is \(c = 0\) and the radius of the interval is \(\delta = 2\). The inequality that represents the interval -2,2 is \(|x| < 2\).

Step 1: Determine the center of the interval (c)

The center of the interval \(-2,2\) is the average of -2 and 2. To calculate the center, the formula we use is \(\frac{-2 + 2}{2}\), which will give us 0. Therefore, \(c = 0\).

Step 2: Determine the radius of the interval (\(\delta\))

The radius of the interval \(-2,2\) is the difference between the upper limit and the center (or the lower limit and the center, as this is symmetrical in the number line). Then, let us calculate \$\delta\$ using the formula \(2 - 0\), which gives us 2. Therefore, \(\delta = 2\).

Step 3: Formulate the inequality

Now that we have determined the center and radius, we can formulate the inequality as follows: \(|x - 0| < 2\), which simplifies to \(|x| < 2\). Therefore, the inequality \(|x - 0| < 2\) represents the interval \(-2,2\).

Key Concepts

Inequality Solutions

When we look at solving inequalities involving absolute values, we aim to find all possible values of the variable that make the inequality true. In simpler terms, we are searching for the solution set. This set can often be represented as a range of numbers, which is either finite (like a closed interval [a, b]), or infinite (such as \(x > a\) or \(x < b\)).

For instance, the equation \( |x - c| < \delta \) is an absolute value inequality that tells us the distance between any number \(x\) and a fixed number \(c\) is less than some positive value \(\delta\). Solving this type of inequality usually entails two steps: one, where \(x\) is greater than \(c\) but less than \(c + \delta\), and another, where \(x\) is less than \(c\) but greater than \(c - \delta\). The solution to \( |x| < 2 \) from our original exercise says that \(x\) can be any number between -2 and 2, not including -2 and 2 themselves, since those would not satisfy the \( < \) condition.

Interval Notation

Interval notation is a method used to represent the set of all possible solutions to an inequality on the number line. It’s a concise way to convey a lot of information. For example, in interval notation, the set of all \(x\) such that \(a < x < b\) is represented as \( (a, b) \).

There are four main types of intervals: open (\( (a, b) \)), closed (\( [a, b] \)), half-open (\( [a, b) \)) or (\( (a, b] \)), and infinite (\( (-\infty, a) \)) or (\( (b, +\infty) \)). Here is what each represents:

• Open intervals \( (a, b) \): \(x\) is greater than \(a\) and less than \(b\), but doesn't include the endpoints \(a\) and \(b\).
• Closed intervals \( [a, b] \): \(x\) is greater than or equal to \(a\) and less than or equal to \(b\), including the endpoints themselves.
• Half-open intervals either include only one of the endpoints \(a\) or \(b\) but not both.
• Infinite intervals extend indefinitely in one direction or the other.

An open interval like \( (-2, 2) \) from our exercise includes all the real numbers between -2 and 2, without the numbers -2 and 2 themselves.

Absolute Value

The absolute value of a real number is its distance from zero on the number line, regardless of the direction. It is denoted as \( |x| \) and is always nonnegative. For instance, the absolute value of both -3 and 3 is 3. This is because they are both 3 units away from zero.

In more formal mathematical terms, the absolute value function is defined as:

• \( |x| = x \) if \(x \geq 0\)
• \( |x| = -x \) if \(x < 0\)

This means that if the number is positive or zero, its absolute value is the same as the number itself. If the number is negative, the absolute value is the number’s opposite (positive).

Understanding absolute value is crucial when solving inequalities because it combines two scenarios: one where the variable is positive and another where it is negative. In our example, the inequality \( |x| < 2 \) can be split into two parts: when \(x\) is positive (\(x < 2\)) and when \(x\) is negative (\(-x < 2\), which simplifies to \(x > -2\)). Both parts must be considered to find the complete solution.