Problem 4. (Option 1)

Let $\min (a ; c)$ denote the smallest of the numbers a and c. Construct the graph of the function $y=\min \left(x+2 ; x^{2}-6 x+8\right), \quad$ and use it to solve the inequality $\min \left(x+2 ; x^{2}-6 x+8\right) \geq 0$.
Solution: 1) Find the points of intersection of the graphs $y=x+2, y=x^{2}-6 x+8$, we get $\mathrm{x}=1$ and $\mathrm{x}=6$: for $x \in(-\infty ; 1) \cup(6 ;+\infty)$ the smallest $y=x+2$; for $x \in[1 ; 6]_{\text {smallest of }}$ the numbers a and c $\left.y=x^{2}-6 x+8 ; 2\right)$ Find the points of intersection of the graphs with the x-axis, we get $\mathrm{x}=-2, \mathrm{x}=2$ and $\mathrm{x}=4$; 3) Determine the solution of the inequality from the graph $x \in[-2 ; 2] \cup[4 ;+\infty)$.

Answer: $x \in[-2 ; 2] \cup[4 ;+\infty)$.

![](https://cdn.mathpix.com/cropped/2024_05_06_3e61d6cf2cabed1a01b3g-3.jpg?height=680&width=517&top_left_y=294&top_left_x=1363)