How to Solve Equations and Inequalities Involving Absolute Value

Solving equations and inequalities involving absolute value requires a clear understanding of the concept and a systematic approach. Absolute value, denoted as |A|, represents the distance of a value A from zero, irrespective of its sign. This article provides a step-by-step guide to solving both absolute value equations and inequalities, emphasizing critical steps and examples for clarity.

Solving Absolute Value Equations

An absolute value equation is an equation where an expression with absolute value is set equal to a number. The standard form is A B, where A is the expression and B is a non-negative number. The key to solving these equations is to recognize that the absolute value can be positive or negative, leading to two separate cases.

Steps to Solve Absolute Value Equations

Set Up Two Cases: One where the expression inside the absolute value is positive, and one where it is negative. Solve Each Case Separately: Solve the equations derived from both cases. Check Your Solutions: Ensure each solution satisfies the original absolute value condition.

Example: Solve 2x - 3 5.

Set up cases: Case 1: 2x - 3 5 → 2x 8 → x 4 Case 2: 2x - 3 ?5 → 2x ?2 → x ?1

Solving Absolute Value Inequalities

An absolute value inequality is an inequality where an expression with absolute value is compared to a number. The general forms are |A| B. The process involves setting up two cases based on the nature of the inequality and combining the results.

Solving |A| B

For |A| Set Up Two Cases: One where A -B. Combine the Results: Typically, this leads to a compound inequality.

Example: Solve |x - 2| Set up cases: Case 1: x - 2 Case 2: x - 2 > -3 → x > -1

Combine: The solution is -1

Solving |A| B

For |A| > B, the steps are as follows:

Set Up Two Cases: One where A > B and one where A Solve Each Case Separately: Solve the inequalities derived from both cases. Combine the Results: This often results in a compound inequality or a union of intervals.

Example: Solve |x - 1| > 2.

Set up cases: Case 1: x - 1 > 2 → x > 3 Case 2: x - 1

Combine: The solution is x 3.

Summary

For equations, set up two cases based on the positive and negative possibilities. For inequalities, consider both the less than and greater than scenarios, which often lead to compound inequalities or unions of intervals. Always check your solutions against the original equations or inequalities.

Feel free to ask for more examples or clarification on specific types of absolute value problems!