# Wavy curve method

Most of us know how to solve equations. But what if you were presented an inequality of the form \(\dfrac{f(x)}{g(x)}\) which is greater than, greater than or equal to, less than, or less than or equal to 0? There are multiple ways to solve this. One way is the **wavy curve method**.

## Basic examples

## Solve \(\dfrac{3x - x^2}{{(x + 4)}^2} \geq 0.\)

Step 1

Factorize the polynomials: \[\begin{align} \dfrac{x(3 - x)}{{(x + 4)}^2} & \geq 0 \end{align}.\]

Step 2

Make the coefficient of the variable of all factors positive: \[\begin{align} \dfrac{-x(x - 3)}{{(x + 4)}^2} & \geq 0 \end{align}.\]

Step 3

Multiply/divide by -1 both sides of the inequality to remove the minus sign (but in doing so, the inequality would reverse): \[\begin{align} \dfrac{x(x - 3)}{{(x + 4)}^2} & \leq 0 \end{align}.\]

Step 4

Find the roots of the inequality by equating each factor to 0: \[\begin{align} x & = 0\\ x - 3 = 0 \implies x & = 3\\ x + 4 = 0 \implies x & = -4. \end{align}\]

Step 5

Plot the points on the number line. Now, start with the largest factor, i.e. 3. Initially, a curve from the positive region of the number line should intersect that point (here 3). Now, look at the power of the respective factors. If it is odd, then we have to change the path of the curve from their respective roots. If it is even, continue in the same region. Here, the curve would change its path at 0 and 3 because their factors are odd powers. However, at 4, it would not change its direction since its factor has an even power.Now, if the inequality is either \(\geq\) or \(\leq\) 0, then we have to consider those values of \(x\) at which the inequality is equal to 0. However, as a rule of the wavy curve method, we should exclude the root of the factor in the denominator (here -4) in our solution set.

So, our final answer is \(x \in [0,3]. \ _\square\)

\[\dfrac{x{(x+5)}^{2016}{(x-3)}^{2017}{(6-x)}^{1231}}{{(x-2)}^{10000}{(x+1)}^{2015}{(4-x)}^{242}} \geq 0\]

Find all the possible values of \(x\).

**Clarification**: In the options, \(\cup \) stands for union.

**Cite as:**Wavy curve method.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/wavy-curve-method/