The Range Finder
Bounds & Error Intervals: Understanding the Hidden Range
🎯The Range Finder
Error Interval: 55 ≤ x < 65
💡The Logic Track
The Half-Unit Rule
When a value is rounded to the nearest 10, the actual value could be anywhere within ±5 of the rounded number.
Closed vs Open Bounds
Lower Bound: 55 (closed ●) - The value CAN equal this.
Upper Bound: 65 (open ○) - The value is LESS THAN this.
⚠️ Examiner Trap
Students often write 55 ≤ x ≤ 65. WRONG! The upper bound uses "<" not "≤".
📋Key Formulas
Error (Half-Unit)
Error = Unit ÷ 2
Lower Bound
LB = Value − Error
Upper Bound
UB = Value + Error
Error Interval
LB ≤ x < UB
⚙️The Dynamic Solver
Step 1: Identify the rounding unit
Value: 60
Rounded to nearest: 10
Step 2: Calculate the error (half-unit rule)
Error = 10 ÷ 2
Error = 5
Step 3: Find the bounds
Lower Bound = 60 − 5
LB = 55
Upper Bound = 60 + 5
UB = 65
Step 4: Write the error interval
⚠️ Common Error: Using "≤" for both bounds. Remember: lower bound is CLOSED (can equal), upper bound is OPEN (cannot equal).
🎓 Exam Strategy: The 3-Phase Bridge
Phase 1: Foundation (Grade 4-5)
Find bounds for single values. Write error intervals correctly with the "≤" and "<" symbols.
Phase 2: Higher (Grade 6-7)
Calculate bounds for products (A × B). Show all four combinations clearly.
Phase 3: Killer (Grade 8-9)
Complex calculations like (A + B) ÷ (C − D). Master the "denominator flip" rule for division.
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