#### Answer

$x = \frac{5}{2}$
$LN = \frac{25}{2}$
$MP = \frac{25}{2}$

#### Work Step by Step

According to Theorem 6-15, the diagonals of a rectangle are congruent. Therefore, we can set $LN$ and $MP$, the diagonals of $LMNP$, equal to one another to solve for $x$:
$LN = MP$
Substitute with the expressions given for each diagonal:
$3x + 5 = 9x - 10$
Subtract $9x$ from each side of the equation to move variables to the left side of the equation:
$-6x + 5 = -10$
Subtract $5$ from each side of the equation to move constants to the right side of the equation:
$-6x = -15$
Divide each side by $-6$ to solve for $x$:
$x = \frac{15}{6}$
Divide the numerator and denominator by their greatest common factor, $3$:
$x = \frac{5}{2}$
Now that we have the value of $x$, we can plug $\frac{5}{2}$ in for $x$:
$LN = 3(\frac{5}{2}) + 5$
Multiply first, according to order of operations:
$LN = \frac{15}{2} + 5$
Convert $5$ into a fraction that has $2$ as its denominator:
$LN = \frac{15}{2} + \frac{10}{2}$
Subtract to solve:
$LN = \frac{25}{2}$
Now let's find $MP$:
$MP = 9(\frac{5}{2}) - 10$
Multiply first, according to order of operations:
$MP = \frac{45}{2} - 10$
Convert $10$ into a fraction with $2$ as its denominator:
$MP = \frac{45}{2} - \frac{20}{2}$
Add to solve:
$MP = \frac{25}{2}$