Wavy Curve Method

The wavy curve method (also called the method of intervals) is a strategy used to solve inequalities of the form $\frac{f(x)}{g(x)} > 0$ $\left(<0, \, \geq 0, \, \text{or} \, \leq 0\right).$ The method uses the fact that $\frac{f(x)}{g(x)}$ can only change sign at its zeroes and vertical asymptotes, so we can use the roots of $f(x)$ and $g(x)$ to sketch a graph of the function over different intervals.

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

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Step 1
Factor the polynomials: $$\begin{aligned} \dfrac{x(3 - x)}{{(x + 4)}^2} & \geq 0 \end{aligned}.$$
Step 2
Make the coefficient of the variable of all factors positive: $$\begin{aligned} \dfrac{-x(x - 3)}{{(x + 4)}^2} & \geq 0 \end{aligned}.$$
Step 3
Multiply/divide both sides of the inequality by -1 to remove the minus sign (remember that in doing so the inequality would reverse): $$\begin{aligned} \dfrac{x(x - 3)}{{(x + 4)}^2} & \leq 0 \end{aligned}.$$
Step 4
Find the roots and asymptotes of the inequality by equating each factor to 0: $$\begin{aligned} x & = 0\\ x - 3 = 0 \implies x & = 3\\ x + 4 = 0 \implies x & = -4. \end{aligned}$$
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 $x \in [0,3]. \ _\square$