Invent an inequality given its solution (I)

Text

Invent an inequality whose solution set is $-3\le x <0 \ \vee\ x \ge 4$.

Solution

Work backwards by finding the breakpoints: $-3$ and $4$ included (zeros of the numerator), $0$ excluded (zero of the denominator). Build a quotient with these breakpoints and adjust the direction by studying its sign: $\frac{(x+3)(x-4)}{x}\ge 0.$ Sign table with breakpoints $-3,0,4$:

• $x<-3$: negative;
• $-3\le x<0$: positive ($-3$ included, $0$ excluded);
• $0<x<4$: negative;
• $x\ge 4$: positive ($4$ included).

The solution set is exactly $-3\le x<0 \ \vee\ x\ge 4$. $\boxed{\frac{(x+3)(x-4)}{x}\ge 0}$ (Any inequality with the same breakpoints and sign behaviour is equally valid.)