Problem 22 - College Algebra

Question

Solve the inequality involving absolute value. Write your final answer in interval notation. $$ |x-20|>-1 $$

Step-by-Step Solution

Answer

The solution is \((-\infty, \infty)\).

Step 1: Understand the Absolute Value Condition

The inequality involves an absolute value, which usually results in two separate inequalities. However, for any real number, the absolute value is always non-negative. Thus, \(|x-20|\geq0\).

Step 2: Simplify Based on Absolute Value Properties

Given the condition \(|x-20| > -1\), recognize that an absolute value, being non-negative, is always greater than any negative number. Thus, this condition is always true for all real numbers.

Step 3: Conclude with Interval Notation

Since there are no restrictions, the solution includes all real numbers. Therefore, in interval notation, the solution is \((-\infty, \infty)\).

Key Concepts

Understanding Interval Notation

Interval notation is a simple way to express a range of numbers. Think of it as a concise version of describing where numbers lie on the number line.
When you see an interval like \(a, b\), it represents all numbers between \(a\) and \(b\), not including \(a\) and \(b\) themselves. If we include these endpoints, we use brackets instead, like this: \[a, b\].

Here's a quick rundown of the symbols:

• "( )" means the endpoints are not included (open interval).
• "[ ]" means the endpoints are included (closed interval).

For example, \((0, 5]\) means all numbers greater than 0 and up to and including 5. In solving inequalities, as we did with \(|x-20|>-1\), understanding how to express the answer in interval notation is crucial. This particular solution, \((-\infty, \infty)\), indicates all real numbers are included, pointing out no restrictions exist.

Grasping Absolute Value Properties

The absolute value of a number \(x\), written as \(|x|\), represents its distance from zero on the number line. It's always non-negative. Two essential properties are helpful in solving inequalities involving absolute values:

• Non-negativity: \(|x| \geq 0\) for any real number \(x\). This property ensures that absolute values are never negative.
• Two-case nature: For a positive number \(c\), \(|x| < c\) translates to \(-c < x < c\). Similarly, \(|x| > c\) splits into \(x > c\) or \(x < -c\).

In the given problem \(|x-20| > -1\), the critical realization is that since \(|x-20|\) must be non-negative, it is always greater than \(-1\). This insight shows that no specific calculations are needed, as any real number satisfies the inequality.

The Concept of Real Numbers

Real numbers encompass all the numbers we typically think of, like whole numbers, fractions, and irrational numbers. They can be located anywhere on the infinite number line, extending in both directions.

When solving inequalities like \(|x-20| > -1\), we often refer to solutions in terms of real numbers. Since absolute values can't be negative, when faced with an inequality that's always true (as \(-1\) is less than any non-negative number), it implies the solution space is vast, covering:

• All positive numbers
• All negative numbers
• The number zero

That is why the solution \((-\infty, \infty)\) translates to all real numbers — because they all satisfy the inequality in this case. This inclusivity highlights how real numbers relate to solving inequalities, showing a complete set without exceptions.