Inequalities involving absolute values are mathematical statements that contain at least one absolute value expression, such as |x+2| or |x2 - 3ax + 2a2|. These expressions represent the distance of a number from zero, and the inequality states that the value of the expression must be less than or greater than a certain number.
To solve inequalities involving absolute values, you must first isolate the absolute value expression on one side of the inequality. Then, you must consider two cases: when the value inside the absolute value is positive and when it is negative. For each case, you can remove the absolute value by applying the appropriate inequality symbols and solve for the variable.
Solving inequalities with one absolute value involves only one absolute value expression, while solving inequalities with two absolute values involves two absolute value expressions. This means that there may be two possible solutions for the variable in the latter case.
The solution set for inequalities involving absolute values is the set of all values of the variable that satisfy the inequality. After solving for the variable, you can graph the solution set on a number line or express it in interval notation.
Inequalities involving absolute values are commonly used in real-life situations involving distance, such as calculating the minimum or maximum distance between two points, or determining the acceptable range of values for a measurement. They are also used in optimization problems, where the goal is to minimize or maximize a certain value.