Absolute value
Methods for solving absolute value tasks
Not every absolute value problem calls for the same method. The best choice depends on the number of absolute values and whether the task reads naturally as distance.
Three key methods
Each method leads to the same result, but they differ in calculation comfort.
Definition method
Use it when you need to formally consider the sign of the expression inside absolute value.
Geometric method
Best for equations and inequalities such as |x - a| = r, |x - a| < r, |x - a| > r.
Splitting into intervals
The most reliable method for many absolute values, because each one can change sign at a different point.
Checking the result
After solving, always check whether the result belongs to the interval in which it was computed.
When should you use each method?
Simple distanceSolve |x - 4| = 2 geometrically: x = 2 or x = 6
One absolute value and x on the rightSolve |x + 1| = 2x by cases from the definition
Several absolute valuesFor |x - 1| + |x + 3| split the line at x = 1 and x = -3
Inequality with a radiusTurn |x - a| < r into a - r < x < a + r
The main rule
First ask whether the task can be read as distance. If yes, the geometric method is shortest. If not, find the zeros of the expressions inside absolute values and consider the resulting intervals.