How to solve (x-9)(x+1) >= 0

1. Factor the equation (x + 2)(x + 3) = 0. 2. Set each factor equal to zero: x + 2 = 0 and x + 3 = 0. 3. Solve for x: x = -2 and x = -3.

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$$(x-9)(x+1) \geq 0$$

$x\in (-\infty,-1]\cup [9,\infty)$

For the product to be $≥0$, $x+1$ and $x-9$ have to be both $≤0$ or both $≥0$. Consider the case when $x+1$ and $x-9$ are both $≤0$.

$$x+1\leq 0$$ $$x-9\leq 0$$

The solution satisfying both inequalities is $x\leq -1$.

$$x\leq -1$$

Consider the case when $x+1$ and $x-9$ are both $≥0$.

$$x-9\geq 0$$ $$x+1\geq 0$$

The solution satisfying both inequalities is $x\geq 9$.

$$x\geq 9$$

The final solution is the union of the obtained solutions.

$$x\leq -1\text{; }x\geq 9$$