Modulus Equations
A modulus equation is solved by removing the bars correctly. For |ax + b| = c, first check c ≥ 0 (if c < 0 there is no solution), then split into ax + b = c and ax + b = −c and solve each. Always substitute solutions back, because squaring or combining moduli can create false roots. The richer CAT type is the sum-of-distances equation such as |x − a| + |x − b| = c. Here the critical points are a and b; they cut the number line into three regions, and in each region every modulus opens with a fixed sign, turning the equation into a simple linear one. A geometric shortcut: |x − a| + |x − b| is the total distance from x to a and to b, whose minimum value is |a − b|. So if c < |a − b| there is no solution; if c = |a − b| every point between a and b works; if c > |a − b| there are exactly two solutions, one on each side.
✅ Solved examples
✏️ Practice — try these, take hints as needed
📝 Topic test — 8 questions
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Formula Reference Sheet
Definition & core properties
| Piecewise definition | |x| = x if x ≥ 0, and −x if x < 0 |
|---|---|
| Square-root form | |x| = √(x²); also |x|² = x² |
| Distance on the line | |x − a| = distance between x and a |
| Product & quotient | |ab| = |a||b|; |a/b| = |a|/|b| (b ≠ 0) |
| Non-negativity | |x| ≥ 0, with |x| = 0 only when x = 0 |
Equations, inequalities & triangle rule
| Basic equation | |x| = c (c ≥ 0) ⇒ x = c or x = −c |
|---|---|
| Linear equation | |ax + b| = c ⇒ ax + b = ±c (needs c ≥ 0) |
| Less-than inequality | |x| ≤ c ⇒ −c ≤ x ≤ c (c ≥ 0) |
| Greater-than inequality | |x| ≥ c ⇒ x ≤ −c or x ≥ c |
| Triangle inequality | |a + b| ≤ |a| + |b|; |a − b| ≥ ||a| − |b|| |