Solve (x+1)(x+6)leq0

$$( x + 1 ) ( x + 6 ) \leq 0$$

Solve for x

$$x\in \begin{bmatrix}-6,-1\end{bmatrix}$$

Solution Steps

For the product to be ≤0, one of the values x+6 and x+1 has to be ≥0 and the other has to be ≤0. Consider the case when x+6\geq 0 and x+1\leq 0.

$$x+6\geq 0 x+1\leq 0$$

The solution satisfying both inequalities is x\in \left[-6,-1\right].

$$x\in \begin{bmatrix}-6,-1\end{bmatrix}$$

Consider the case when x+6\leq 0 and x+1\geq 0.

$$x+1\geq 0 x+6\leq 0$$

This is false for any x.

$$x\in \emptyset$$

The final solution is the union of the obtained solutions.

$$x\in \begin{bmatrix}-6,-1\end{bmatrix}$$