Absolute value functions, equations, and inequalities

The absolute value of a number is its distance from 0, so |x| is never negative. The graph of f(x) = |x| is a V shape with its vertex at (0, 0), and f(x) = a|x - h| + k moves the vertex to (h, k), with a stretching the V and a negative a flipping it upside down, while f(bx) compresses or stretches it horizontally. Absolute value equations model distance: the numbers 3 units from 12 satisfy |x - 12| = 3, and a part that must be within 0.5 mm of 20 mm satisfies |x - 20| <= 0.5. To solve an absolute value equation, isolate the absolute value, then split it into two cases: |x - 3| = 5 means x - 3 = 5 or x - 3 = -5, so x = 8 or x = -2. An equation like |2x + 1| = -4 has no solution, since an absolute value cannot be negative. For inequalities, a "less than" becomes a compound "and" inequality between two values: |x - 2| <= 3 means -3 <= x - 2 <= 3, so -1 <= x <= 5. A "greater than" becomes an "or": |x| > 4 means x < -4 or x > 4.

This explanation is AI-generated and can occasionally be wrong. 🚩 Report