4. All passengers of the small cruise liner "Victory" can be accommodated in 7- and 11-seat lifeboats in case of emergency evacuation, with the number of 11-seat lifeboats being greater than the number of 7-seat lifeboats. If the number of 11-seat lifeboats is doubled, the total number of lifeboats will be more than 25, and if the number of 7-seat lifeboats is doubled, the total number of lifeboats will be less than 29. Determine the number of passengers on the cruise liner "Victory".

(7 points)
Answer: 60 possible options are given in the table below.

## Solution:

Let $x, y$ be the number of 7-seater and 11-seater boats, respectively, and $z$ be the total number of passengers.

Then $z=7 x+11 y$, where $x, y$ satisfy the system of inequalities: $\left\{\begin{array}{c}2 y+x>25, \\ 2 x+yx\end{array}\right.$ Let's consider the first two inequalities of the system in more detail.

1) Multiplying the first inequality by (-2), we get $\left\{\begin{array}{c}-4 y-2 x7\end{array}\right.$.
2) Multiplying the second inequality by (-2), we get $\left\{\begin{array}{c}2 y+x>25, \\ -4 x-2 y>-58\end{array}\right.$. Adding these two inequalities, we get $-3 x>-33$, hence, $\boldsymbol{x}<11$.

Next, we perform a selection that satisfies the system of inequalities and the obtained constraints, and then calculate $z=7 x+11 y$ - the total number of passengers (the values of $z$ are located in the cells at the intersection of rows and columns for the corresponding $x, y$ ).

| $\underline{x}$ |  | $\sqrt{8}$ | 7 | 6 | 5 | 4 | $\sqrt{3}$ | $\sqrt{2}$ | $\\| 1$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $\mathbf{8}_{8}$ |  |  |  |  |  |  |  |  |  |
|  |  | $\|155\|$ |  |  |  |  |  |  |  |
| 10 | 173 | 166 | \|159| | 152 | 156 |  |  |  |  |
| 11 |  | 177[ | 170\| | 163 | 167 | \|149| |  |  |  |
| 12 |  | 188] | 181\| | 174 | 178 | \|160| | \|153| | 146 |  |
| 13 |  | 192\| | 185 | 189 | \|171| | \|164| | 157 | 150 |
| 14 |  |  | 203[ | 196 | 200 | \|182| | \|175| | 168 | 161 |
| 15 |  |  | 207 | 211 | \|193| | \|186| | 179 | 172 |
| 16 |  |  |  | 218 | 222 | 204 | \|197| | 190 | 183 |
| $\overline{17}$ |  |  |  |  | 233 | 215\| | 208 | 201 | $\sqrt{192}$ |
| 18 |  |  |  |  |  | 226 | \|219| | 212 | 205 |
| 19 |  |  |  |  |  | 237 | 230 | 223 | 216 |
| 20 |  |  |  |  |  | 248 | 241 | 234 | $\sqrt{227}$ |
| 21 |  |  |  |  |  |  | 2252 | 245 | 238 |
| 22 |  |  |  |  |  |  | \|263| | 256 | 249 |
| 23 |  |  |  |  |  |  |  | 267 | 260 |
| 24 |  |  |  |  |  |  |  | $[278$ | 271 |
| 25 |  |  |  |  |  |  |  |  | 282 |
| \|26 |  |  |  |  |  |  |  |  | 293 |