Solving Polynomial Inequalities with the Wavy Curve Method

The Wavy Curve Method: A Step-by-Step Approach

The core idea of the wavy curve method is to find the points where an expression can change its sign—these are its roots (where it equals zero) and its points of discontinuity (where the denominator is zero). These points, called critical points, divide the number line into intervals. Within each interval, the sign of the expression remains constant.

The following resource from BYJU'S provides a clear, algorithmic breakdown of this method.

Let's illustrate with the first solved example from the same article.
Problem: For $P(x)=\frac{(x-3)(x+2)}{(x-7)(x+1)}$, find where $P(x)$ is positive or negative.

1. Find Critical Points: The roots of the numerator are $3$ and $-2$. The roots of the denominator are $7$ and $-1$. These are the points where the sign can change.
2. Order on Number Line: Arrange them in increasing order: $-2,-1,3,7$.
3. Draw the Wavy Curve: Start from the far right (for $x>7$). Since all factors are of the form $(x-a)$ and have positive coefficients of $x$, the expression will be positive. Draw the curve starting from above the axis. At each critical point, cross the axis (assuming all factors have odd powers, which they do here—power 1).

From the diagram, we can read the solution:

• $P(x)>0$ in the intervals $(-\infty ,-2)\cup (-1,3)\cup (7,\infty )$.
• $P(x)<0$ in the intervals $(-2,-1)\cup (3,7)$.

Important Note on 'Greater/Less Than or Equal To' ($\ge ,\le$):
When dealing with non-strict inequalities:

• Roots from the numerator are included in the solution set (closed circles, []).
• Roots from the denominator are always excluded (open circles, ()), as they make the expression undefined.

Advanced Cases and JEE Problem Patterns

For JEE Advanced, you'll encounter more complex inequalities involving repeated factors (even/odd powers), square roots, logarithms, and other functions. The underlying wavy curve logic remains the same, but you must handle these special cases correctly.

The following video from Vedantu JEE works through a Menti quiz that covers many of these essential variations and common pitfalls. It’s highly practical and aligns with your preference for problem-based learning.

Practice Problems

Now it's time to apply these techniques yourself. The following resource is a Daily Practice Problem (DPP) sheet. Try to solve the problems first, then check your answers against the provided solutions.

Conclusion

In this lesson, you've mastered the wavy curve method, a fundamental technique for solving polynomial and rational inequalities.

Key Takeaways:

• The method relies on finding critical points (roots and discontinuities) to divide the number line into intervals of constant sign.
• The sign of the expression crosses the axis at roots with odd multiplicity and bounces off (retains its sign) at roots with even multiplicity.
• Always ensure the expression is in factored form with the coefficient of $x$ being positive in each factor. If not, factor out a negative and reverse the inequality sign if necessary.
• Be vigilant about the domain of the expression, especially for square roots and logarithms. An inequality is only valid where the function is defined.
• For $\ge$ or $\le$, roots of the numerator are included, while roots of the denominator are always excluded.

In our next lesson, we will build upon these concepts to solve problems related to the location of roots of a quadratic equation. This involves finding conditions on the coefficients of a quadratic so that its roots lie in a certain interval, a classic problem type for JEE Advanced that merges inequalities with the properties of parabolas.