Find the range of values of m which satisfy (m - 3)(m - 4) < 0

Analyze the question: We need the values of m for which the product (m − 3)(m − 4) is negative (less than zero).

Concept: A product of two factors is negative only when the factors have opposite signs. For a quadratic inequality, find the critical points (the roots) first, then test the regions between and outside them.

Solution:

1. Find the critical points by setting each factor to zero: • m − 3 = 0 gives m = 3 • m − 4 = 0 gives m = 4

2. These divide the number line into three regions. Test each: • m < 3 (try m = 0): (−3)(−4) = 12, which is positive ✗ • 3 < m < 4 (try m = 3.5): (0.5)(−0.5) = −0.25, which is negative ✓ • m > 4 (try m = 5): (2)(1) = 2, which is positive ✗

3. The inequality holds only when 3 < m < 4.

Check with the sign rule:

• Case 1: m − 3 > 0 and m − 4 < 0 gives m > 3 and m < 4, so 3 < m < 4 ✓
• Case 2: m − 3 < 0 and m − 4 > 0 gives m < 3 and m > 4, which is impossible.

Answer: C. 3 < m < 4

Exam tip: For (x − a)(x − b) < 0 with a < b, the answer is always between the roots: a < x < b. For (x − a)(x − b) > 0, the answer is outside the roots: x < a or x > b. Think of the graph: an upward-opening parabola dips below the x-axis between its roots.