Solve each equation or inequality. | 12- 9x | ≥ -12

Verified step by step guidance

1. Recognize that the inequality involves an absolute value: \(|12 - 9x| \geq -12\).
2. Understand that an absolute value expression is always non-negative, meaning \(|12 - 9x|\) is always greater than or equal to 0.
3. Since \(-12\) is less than 0, the inequality \(|12 - 9x| \geq -12\) is always true for any real number \(x\).
4. Conclude that the solution to the inequality is all real numbers, because the absolute value expression is always greater than or equal to any negative number.
5. Express the solution set as \(x \in \mathbb{R}\), indicating that any real number satisfies the inequality.

Key Concepts

Absolute Value

Absolute value measures the distance of a number from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|, and is always non-negative. For example, |3| = 3 and |-3| = 3. Understanding absolute value is crucial for solving equations and inequalities that involve it, as it can lead to two separate cases based on the definition.

Inequalities

Inequalities express a relationship between two expressions that are not necessarily equal, using symbols like ≥, ≤, >, or <. In this context, the inequality |12 - 9x| ≥ -12 indicates that the expression inside the absolute value is either greater than or equal to -12 or less than or equal to 12. Recognizing how to manipulate and solve inequalities is essential for finding valid solutions.

Case Analysis

When solving absolute value equations or inequalities, case analysis is a method used to break down the problem into simpler parts. For the inequality |12 - 9x| ≥ -12, we consider the two scenarios: when the expression inside the absolute value is non-negative and when it is negative. This approach allows us to solve for x in a structured manner, ensuring all possible solutions are considered.