General Approach to Solving Inequalities Involving Absolute Values

Inequalities involving absolute values are encountered frequently in both mathematics and practical applications. To solve these inequalities, one often employs the concept of the absolute value function and breaks the problem into smaller, manageable pieces using a piecewise approach. This article outlines a methodical and comprehensive strategy for solving such inequalities, providing examples and explanations.

Introduction to the Absolute Value Function

The absolute value function, denoted as |x|, is defined as follows:

begin{equation}|x| begin{cases} x text{if } x geq 0 -x text{if } x

For instance, when dealing with an inequality of the form |f(x)| leq a, where (a > 0), it can be converted into two separate inequalities:

begin{equation}-f(x) leq a leq f(x)end{equation}

This can be broken down into:

begin{equation}begin{cases}f(x) geq -a f(x) leq aend{cases}end{equation}

These two inequalities can then be solved individually to find the solution set.

Example: Solving (|x - 1| leq 1)

Let's consider an example to illustrate the process. We want to solve the inequality:

[|x - 1| leq 1]

First, we express (|x - 1|) as a piecewise function:

[|x - 1| begin{cases} 1 - (x - 1) 2 - x text{if } x This gives us two inequalities to solve:

For (x For (x geq 1): (x - 1 leq 1)

Solving these individually:

For (x

For (x geq 1): (x - 1 leq 1) simplifies to (x leq 2). Thus, the solution for this part is (1 leq x leq 2).

Combining these results, the solution set is: (1 leq x leq 2).

Example: Solving (|2x - 1| geq 1)

Let's provide another example, which involves an inequality where the absolute value is greater than or equal to a constant:

[|2x - 1| geq 1]

This can be broken down into two cases:

begin{equation}begin{cases}2x - 1 geq 1 2x - 1 leq -1end{cases}end{equation}

Solving these inequalities:

(2x - 1 geq 1) simplifies to (2x geq 2), hence (x geq 1).

(2x - 1 leq -1) simplifies to (2x leq 0), hence (x leq 0).

Combining these results, the solution set is: (x leq 0) or (x geq 1).

Conclusion

In summary, solving inequalities involving absolute values often requires breaking the problem into smaller, piecewise-defined functions. By handling each case separately and combining the results, we can find the complete solution set. This approach is systematic and can be applied to a wide range of absolute value inequalities.

Keywords

absolute value inequality, piecewise function, algebraic solution