Absolute Value Equations and Inequalities
There are three types of absolute value equations and inequalities. The key to solving these is identifying which type you're looking at so you can identify the right procedure. In all three cases, the procedure only applies when $c\ge0$.
| Type of Equation/Inequality | Solution Method |
| $$|x|=c$$ | $$x=c \text{ or } x=-3$$ |
| $$|x|>c$$ $$|x|\ge c$$ | $$-c < x < c$$ $$-c \le x \le c$$ |
| $$|x| < c$$ $$|x|\le c$$ | $$x < c \text{ or } x > -c$$ $$x \le c \text{ or } x \ge -c$$ |
Notice that the only difference for < versus $\le$ and > versus $\ge$ is the "equal" part of the inequality.
Video Lectures
Lectures
Equations in the form |ax + b| = c can be solved using a simple procedure assuming c is non-negative. (lecture slides)
The method for solving absolute value inequalities comes in two forms depending on the direction of the inequality. (lecture slides)
The methods for solving absolute value inequalities and equations that we've discussed so far all require the constant value to be non-negative. In this lecture, we'll look at how to handle each type of equation or inequality when the constant value is negative. (lecture slides)
