#### Answer

This compound inequality has $\textbf{no solution}$.

#### Work Step by Step

$5-x\gt7$ and $2x+3\ge13$
Solve the first inequality:
$5-x\gt7$
Take $x$ to the right side and $7$ to the left side:
$5-7\gt x$
$-2\gt x$
Rearrange:
$x\lt-2$
Expressing the solution in interval notation:
$(-\infty,-2)$
Solve the second inequality:
$2x+3\ge13$
Take $3$ to the right side:
$2x\ge13-3$
$2x\ge10$
Take $2$ to divide the right side:
$x\ge\dfrac{10}{2}$
$x\ge5$
Expressing the solution in interval notation:
$[5,\infty)$
Since the compound inequality is formed by the word "and", the solution is composed by the numbers that satisfy both inequalities simultaneously.
The solution would be $(-\infty,-2)\cap[5,\infty)$, but since these two intervals have no elements in common, this compound inequality has no solution.