Question
Find the values of satisfying .
Solution — Step by Step
We need , i.e., the product must be negative.
The expression equals zero at and .
These are the critical points that divide the number line into three intervals:
We check the sign of in each interval:
| Interval | Product | ||
|---|---|---|---|
| (test ) | negative | negative | positive |
| (test ) | positive | negative | negative |
| (test ) | positive | positive | positive |
The product is negative only in the interval .
The endpoints and are excluded (strict inequality — the expression equals 0 there, not less than 0).
Why This Works
A product of two factors is negative when exactly one factor is negative (opposite signs). The factors and change signs at and respectively. Between these roots, but , giving a negative product.
The parabola opens upward (positive leading coefficient). It’s negative between its two roots. This geometric interpretation directly confirms .
Alternative Method — Number Line / Wavy Curve Method
Plot the critical points 2 and 3 on a number line. For a quadratic , the parabola opens upward, so the expression is negative between the roots and positive outside the roots.
Since we want : solution is between the roots → .
The Wavy Curve Method (or Sign Change Method) is much faster for higher-degree inequalities. Mark the zeros of each factor on a number line. To the extreme right, the expression is always positive (for leading positive coefficient). Alternate signs as you cross each zero. This avoids evaluating test points for every interval.
Common Mistake
Students sometimes write the solution as or — exactly the opposite of the correct answer. This confusion arises from thinking “the roots are 2 and 3, so the answer involves these boundary values in the inequalities or .” Remember: for (expression is negative or zero), the solution is between the roots for an upward-opening parabola. For , it’s outside the roots. Verify by substituting — you get ✓.