Problem 100 - Algebra for College Students

Question

Solve each equation or inequality. $$ |x| \geq-15 $$

Step-by-Step Solution

Answer

All real numbers satisfy the inequality.

Step 1: Understand Absolute Value Concept

Absolute value of a number represents its distance from zero on the number line, regardless of the direction. Hence, it is always non-negative. For any value of x, \(|x| \geq 0\).

Step 2: Analyze the Inequality

Given \(|x| \geq -15\), note that the right-hand side (-15) is negative. Absolute values are always non-negative, so any real number will satisfy this inequality.

Step 3: State the Solution

Since all real numbers satisfy \(|x| \geq -15\), the solution is \(x \in \mathbb{R}\).

Key Concepts

Absolute Value

Absolute value is a key concept in mathematics that determines how far a number is from zero on the number line. It does not take direction into account, which means it doesn't matter if the number is positive or negative. The absolute value of any real number is always non-negative.

For example:

• The absolute value of 5 is 5.
• The absolute value of -5 is also 5.
• The absolute value of 0 is 0.

The mathematical representation for absolute value is \(|x|\). This means that if x is a positive number or zero, \(|x|\) is simply x. If x is negative, \(|x|\) is -x. Understanding this foundation is crucial for solving absolute value inequalities.

Inequality

Inequality is similar to an equation, but instead of showing that two expressions are equal, it shows that one expression is either larger or smaller than another. Inequalities use symbols like \(<\), \(>\), \( \leq \), and \( \geq \).

In this exercise, the inequality is \(|x| \geq -15\). Because absolute values are always non-negative (greater than or equal to 0), this inequality is naturally satisfied. A non-negative number will always be greater than or equal to any negative number. Therefore, every real number is a solution to the inequality.

Real Numbers

Real numbers include all positive numbers, negative numbers, and zero. They encompass all the numbers that can be represented on the number line.

When we solve the inequality \(|x| \geq -15\), we determine that all real numbers satisfy this condition. This is why the final solution can be written as \(x \in \mathbb{R}\), indicating that x can be any real number. Understanding the set of real numbers is fundamental to grasping why such inequalities hold true for all x.