INEQUALITIES WITH MODULUS
PAGE 1 OF 5, THE SAFE ZONE: LESS THAN VS GREATER THAN
MODULUS INEQUALITY
FINDING THE SAFE ZONE
A modulus inequality asks: for which x does the absolute value stay within limits? Unlike equations that pin x to exact points, inequalities describe a region or range. The two core patterns are |x| < k (x stays close to zero) and |x| > k (x sits far from zero). Think of k as a boundary fence. Values inside the fence satisfy "less than," and values outside satisfy "greater than." Every modulus inequality reduces to one of these shapes, possibly after you simplify the inside expression.
⚡ CORE RULE
Less than means between: |x| < k ⇔ −k < x < k. Greater than means outside: |x| > k ⇔ x < −k or x > k.
ZONE!
|x| < k
−k < x < k
📍 Open interval between
✅ e.g. |x| < 4 → (−4, 4)
📍 Open interval between
✅ e.g. |x| < 4 → (−4, 4)
|x| > k
x < −k or x > k
📍 Two outer rays
✅ e.g. |x| > 4 → (−∞, −4) ∪ (4, ∞)
📍 Two outer rays
✅ e.g. |x| > 4 → (−∞, −4) ∪ (4, ∞)