11.8 Find the values of $x$ for which all values of the function $y=x^{2}+5 x+6$ belong to the interval $[6,12]$.
11.8 Method I. According to the condition, $6 \leq x^{2}+5 x+6 \leq 12$. Thus, we have the system of inequalities

$$
\left\{\begin{array} { l } 
{ x ^ { 2 } + 5 x \geq 0 } \\
{ x ^ { 2 } + 5 x - 6 \leq 0 }
\end{array} \Rightarrow \left\{\begin{array}{l}
x \leq-5 ; x \geq 0 \\
-6 \leq x \leq 1
\end{array}\right.\right.
$$

As a result, we get the answer: $-6 \leq x \leq-5$ and $0 \leq x \leq 1$.

Method II. Let's construct the parabola $y=x^{2}+5 x+6$ and the lines $y=6$ and $y=12$ (Fig. 11.6). Using the graph, we establish that the values of the function $y=x^{2}+5 x+6$ are within the interval $[6,12]$ for $-6 \leq x \leq-5$ and $0 \leq x \leq 1$.

![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-348.jpg?height=508&width=402&top_left_y=123&top_left_x=749)

Fig. 11.6

Answer: $-6 \leq x \leq-5$ and $0 \leq x \leq 1$.