|x - 2| < 3

Definition

The expression |x - 2| < 3 represents an inequality involving absolute value, indicating that the distance between the variable x and the number 2 is less than 3. This means that x can take values that are within 3 units of 2 on the number line, creating an interval that captures all such numbers. Understanding this concept is crucial for working with intervals and inequalities, as it helps in visualizing how absolute value defines distances and ranges on the real number line.

5 Must Know Facts For Your Next Test

1. The inequality |x - 2| < 3 can be rewritten as two separate inequalities: -3 < x - 2 < 3.
2. By solving the rewritten inequalities, we find that x must be in the range (−1, 5), which means all values greater than -1 and less than 5 satisfy the original inequality.
3. This expression illustrates how absolute value can describe a set of values around a central point, in this case, 2.
4. The solution set for |x - 2| < 3 can be represented graphically on a number line as an open interval between -1 and 5.
5. Understanding absolute value inequalities like |x - 2| < 3 is essential for solving real-world problems where ranges of values are necessary.

Review Questions

• How would you rewrite the inequality |x - 2| < 3 as a compound inequality, and what does this reveal about the possible values of x?
  • To rewrite |x - 2| < 3 as a compound inequality, you express it as -3 < x - 2 < 3. This reveals that x must be greater than -1 and less than 5, effectively defining an open interval where all numbers within this range satisfy the original inequality. This breakdown helps in understanding how absolute values work in defining distance from a point on the number line.
• Discuss how the concept of intervals relates to the solution of |x - 2| < 3 and why it’s important for understanding ranges of values.
  • The concept of intervals is directly related to solving |x - 2| < 3 since the solution results in an interval of possible x values. Specifically, it results in the open interval (-1, 5). Understanding intervals allows us to visualize and communicate ranges where certain conditions hold true. It is important because many mathematical problems involve determining where functions meet specific criteria, making intervals crucial for solutions.
• Evaluate how understanding absolute value inequalities like |x - 2| < 3 can apply to real-world scenarios and problem-solving.
  • Understanding absolute value inequalities like |x - 2| < 3 can be applied to various real-world scenarios such as determining acceptable ranges for measurements or tolerances in manufacturing. For example, if a product needs to measure close to a standard size (like 2 units), knowing how much deviation is permissible (within ±3 units) helps ensure quality control. This ability to translate mathematical concepts into practical applications shows the importance of mastering these ideas in both academic and professional contexts.

Related terms

Absolute Value:

The absolute value of a number is its distance from zero on the number line, regardless of direction, denoted as |x|.

Interval Notation: A mathematical notation used to represent a range of values on the real number line, typically expressed using brackets or parentheses.

Inequality: An inequality is a mathematical statement that indicates the relationship between two expressions that are not equal, often involving symbols like <, >, ≤, or ≥.