Solve (x+5)(x+1)leq0

$$( x + 5 ) ( x + 1 ) \leq 0$$

Solve for x

$$x\in \begin{bmatrix}-5,-1\end{bmatrix}$$

Solution Steps

For the product to be ≤0, one of the values x+5 and x+1 has to be ≥0 and the other has to be ≤0. Consider the case when x+5\geq 0 and x+1\leq 0.

$$x+5\geq 0 x+1\leq 0$$

The solution satisfying both inequalities is x\in \left[-5,-1\right].

$$x\in \begin{bmatrix}-5,-1\end{bmatrix}$$

Consider the case when x+5\leq 0 and x+1\geq 0.

$$x+1\geq 0 x+5\leq 0$$

This is false for any x.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in \begin{bmatrix}-5,-1\end{bmatrix}$$