Solve (x-3)(x+5)leq0

$$( x - 3 ) ( x + 5 ) \leq 0$$

Solve for x

$$x\in \begin{bmatrix}-5,3\end{bmatrix}$$

Solution Steps

For the product to be ≤0, one of the values x+5 and x-3 has to be ≥0 and the other has to be ≤0. Consider the case when x+5\geq 0 and x-3\leq 0.

$$x+5\geq 0 x-3\leq 0$$

The solution satisfying both inequalities is x\in \left[-5,3\right].

$$x\in \begin{bmatrix}-5,3\end{bmatrix}$$

Consider the case when x+5\leq 0 and x-3\geq 0.

$$x-3\geq 0 x+5\leq 0$$

This is false for any x.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in \begin{bmatrix}-5,3\end{bmatrix}$$