Linear Absolute Value Equations

Absolute value measures distance from zero, so |x-3| means the distance between x and 3 on the number line. When you set that distance equal to a positive number, say |x-3|=5, two points satisfy it: one 5 units to the right of 3, one 5 units to the left. That is why a single absolute value equation hides two ordinary equations: x-3=5 and x-3=-5. Drop the absolute value bars and write both cases, with the second case negating the other side.

Before splitting, isolate the absolute value expression completely, same as you would isolate a squared term. If the equation reads 3|x+4|-7=11, add 7 then divide by 3 to get |x+4|=6 before you split into x+4=6 and x+4=-6. Skipping this step is the single biggest source of wrong answers. Once isolated, check the number on the other side: if it is negative, stop. A distance can never be negative, so an equation like |x-3|=-2 has no solution at all, no matter what case work you try.

This topic is the simplest member of a family that includes absolute value inequalities and absolute value functions graphed as V-shapes. The same case-splitting logic carries forward: inequalities split into two inequalities with a flipped sign on one branch, and the vertex of the graph sits exactly where the expression inside the bars equals zero. Getting the linear equation case solid, including the no-solution check, sets up everything that follows.