Problem 4. (Option 2)

Let $\max (a ; c)$ denote the smaller of the numbers a and c. Construct the graph of the function $y=\max \left(x-11 ; x^{2}-8 x+7\right)$, and use it to solve the inequality $\max \left(x-11 ; x^{2}-8 x+7\right) \prec 0$.

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# Solution:

1) Find the points of intersection of the graphs \( y = x - 11 \) and \( y = x^2 - 8x + 7 \), we get \( x = 3 \) and \( x = 6 \): for \( x \in (-\infty; 3) \cup (6; +\infty) \), the largest \( y = x^2 - 8x + 7 \); for \( x \in [3; 6] \), the largest \( y = x - 11 \);
2) Find the points of intersection of the graphs with the x-axis, we get \( x = 1 \) and \( x = 7 \);
3) Determine the solution of the inequality from the graph, we get \( x \in (1; 7) \).

![](https://cdn.mathpix.com/cropped/2024_05_06_3e61d6cf2cabed1a01b3g-3.jpg?height=657&width=554&top_left_y=1445&top_left_x=1345)

Answer: \( x \in (1; 7) \).

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