An absolute value inequality is an equation that contains an absolute value expression and a relational operator, such as <, >, ≤, or ≥. It represents a range of values that satisfy the inequality, rather than a single solution.
To solve an absolute value inequality, isolate the absolute value expression on one side of the equation and remove the absolute value bars. Then, create two separate equations, one with a positive number on the other side of the inequality sign, and one with a negative number. Solve each equation separately and combine the solutions to find the range of values that satisfy the inequality.
Solving absolute value equations means finding the specific value or values that make the equation true, while solving absolute value inequalities means finding a range of values that make the inequality true. This is because inequalities represent a larger set of solutions than equations.
Yes, absolute value inequalities can have multiple solutions. This is because the absolute value expression can result in a positive or negative value, and each value can satisfy the inequality. Therefore, it is important to check all solutions to ensure they are valid.
Absolute value inequalities can be used to represent a range of values in real-life situations, such as measuring temperatures, distances, or prices. They can also be used in optimization problems, where a certain value must be within a specific range. For example, a company may need to keep their production costs within a certain range to be profitable.