An absolute value inequality is an inequality that includes an absolute value expression. In simpler terms, it's an inequality containing an absolute value symbol.
Understanding Absolute Value Inequalities
Absolute value inequalities are inequalities in algebra that involve algebraic expressions with absolute value symbols and inequality symbols (like <, >, ≤, or ≥). According to the reference, "absolute value inequalities are inequalities in algebra that involve algebraic expressions with absolute value symbols and inequality symbols. In simple words, we can say that an absolute value inequality is an inequality with an absolute value symbol in it."
Examples of Absolute Value Inequalities
Here are some examples to illustrate what absolute value inequalities look like:
- |x| < 5
- |2y + 1| ≥ 3
- |z - 4| ≤ 2
- |3a - 2| > 7
Solving Absolute Value Inequalities
Solving these inequalities involves considering two separate cases due to the nature of absolute value:
- The positive case: Where the expression inside the absolute value is positive or zero.
- The negative case: Where the expression inside the absolute value is negative.
General Forms and Solutions
- |x| < a (where a > 0): This is equivalent to -a < x < a. The solution set is all values of x between -a and a.
- |x| > a (where a > 0): This is equivalent to x < -a or x > a. The solution set is all values of x less than -a or greater than a.
- The same principles apply when using "≤" and "≥" symbols.
