Solve |ax + b| < c (AND case → bounded interval) and |ax + b| > c (OR case → unbounded rays) with step-by-step work and number line visualization. Handles all special cases including no solution and all real numbers.
|expr| < c → −c < expr < c (AND, bounded)
|expr| > c → expr < −c OR expr > c (OR, unbounded)
Less than (< or ≤): |ax + b| < c means ax + b is within c units of 0 — it produces a three-part AND inequality: −c < ax + b < c. Solve all three parts simultaneously. Result: a bounded interval between two values.
Greater than (> or ≥): |ax + b| > c means ax + b is more than c units from 0 — it splits into two separate inequalities: ax + b < −c OR ax + b > c. Solve each branch. Result: two unbounded rays extending to ±∞.
Remember: when dividing by a negative coefficient, flip the inequality sign(s).
|expr| < 0 → No Solution | |expr| > −1 → All Reals
c < 0, less-than type: |expr| < c is impossible — absolute value is always ≥ 0, so it can never be less than a negative number. Answer: No Solution (∅).
c < 0, greater-than type: |expr| > c is always true — any absolute value is greater than every negative number. Answer: All Real Numbers (−∞, +∞).
c = 0, less-than strict: |expr| < 0 is impossible. Answer: No Solution.
c = 0, less-than-or-equal: |expr| ≤ 0 forces expr = 0, giving exactly one point x = −b/a.
c = 0, greater-than strict: |expr| > 0 means expr ≠ 0, so x ≠ −b/a — all reals except one point.
c = 0, greater-than-or-equal: |expr| ≥ 0 is always true. Answer: All Real Numbers.
Work one-on-one with a tutor who breaks down every step clearly — at your pace, on your schedule.