Invent an inequality given its solution (II)

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Invent an inequality whose solution set is $(x \le -3 \ \vee\ 0< x \le 4)\wedge x\neq 3$.

Solution

The required breakpoints are: $-3$ and $4$ included (zeros of the numerator), $0$ excluded (zero of the denominator) and $3$ excluded (it must be “punched out” without changing sign). A factor $(x-3)^2$ in the denominator makes the inequality undefined at $3$, removing it from the domain while preserving the sign. $\frac{(x+3)(x-4)}{x(x-3)^2}\le 0.$ Since $(x-3)^2>0$ for $x\neq 3$, the sign matches that of $\dfrac{(x+3)(x-4)}{x}$, giving $x\le -3 \ \vee\ 0<x\le 4$; removing $x=3$ from the domain yields the condition $x\neq 3$. $\boxed{\frac{(x+3)(x-4)}{x(x-3)^2}\le 0}$ (Any inequality with the same breakpoints and sign behaviour is equally valid.)