Study Guide: Solving Absolute Value Inequalities
Solving Absolute Value Inequalities
Graphing Absolute Value Inequalities
Special Cases of Absolute Value Inequalities
Example 1: All Real Numbers
$$ | 2 - 5x | \ge -4 $$
Remember that absolute value is always either $0$ or a positive number. Because all such numbers are greater than $-4$, all real values for $x$ are true.
Solution: “All real numbers.” This can also be written as $\mathbb{R}$.
Example 2: No Solution
$$ | 4x - 5 | \le -2 $$
Remember that absolute value is always either $0$ or a positive number. Because no such number is less than $-2$, no real values for $x$ can be true.
Solution: “No solution.” This can also be written as $ \emptyset $.
Example 3: Simplify First!
It is important to isolate the absolute value on one side of the inequality before analyzing it. In this example, the inequality symbol is revered when we divide by -2.
\begin{align} -2|4x-5| &\le -8 &\textit{At first glance the solution looks like } \emptyset\textit{.} \\[3ex] \frac{-2|4x - 5|}{{\color{red}-2}} &{\color{red}\ge} \frac{-8}{{\color{red}-2}} & \textit{The inequality symbol reverses. } \\[3ex] |4x-5| &\ge 4 &\textit{Solution: All real numbers, or } \mathbb{R}\textit{.} \end{align}
Conclusions
- If the inequality shows that the absolute value is less than zero or any negative number, there is no solution, $ \emptyset $.
- If the inequality shows that the absolute value is greater than a negative number, the solution set is “All real numbers”, $ \mathbb{R} $.