To solve for $x$, we use the fact that the sum of the interior angles in any quadrilateral is $360^\circ$.

So, add the given angles:

$(10x + 6) + (13x - 2) + (8x - 1) + 71 = 360$

Combine like terms:

$31x + 74 = 360$

Subtract 74 from both sides:

$31x = 286$

Divide both sides by 31:

$x = \frac{286}{31} = 9.23 \, (\text{rounded to two decimal places})$

Hence, $x \approx 9.23$.