Quantity Comparison (Q1 vs Q2)

Master Quantity Comparison (Q1 vs Q2). Solve two independent quantities and compare them ($>, <, =, \ge, \le$). Covers Quadratic, Arithmetic, and Geometry.

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Model 1: The Comparison Framework

  • The Core: You are given two quantities, Quantity I and Quantity II. You must find their values and compare them.
  • Options Standard:
    A) Quantity I > Quantity II
    B) Quantity I < Quantity II
    C) Quantity I тЙе Quantity II
    D) Quantity I тЙд Quantity II
    E) Quantity I = Quantity II or No Relation
  • Strategy: Do NOT estimate unless gap is huge. Calculate exact values. If Q1 depends on a variable range (e.g., x2=4ox=pm2x^2=4 o x=pm2), test ALL cases against Q2.

Example:

Q: Q1: Value of x if x2=25x^2=25. Q2: Value of y if y3=125y^3=125.
Solution: Q1: x=+5,тИТ5x = +5, -5. Q2: y=5y = 5. Compare: +5=5, -5<5. So xleyx le y.

Model 2: Quadratic Inequalities

  • Sign Method: Use sign of constants to predict root signs. (+,+) roots < (-,+) roots? No, verify values.
  • Range Comparison: If Q1 is an interval [a, b] and Q2 is [c, d], compare endpoints. Overlap usually means 'No Relation'.
  • Common Trap: sqrtx2=тИгxтИгsqrt{x^2} = |x| (always +ve), but if x2=16x^2=16, x=pm4x=pm 4. Watch out for these definitions.
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