The problem asks to solve for $x$ given the expressions for the measures of the angles in a quadrilateral inscribed in a circle, where two angles are $3x+11$ and $6x-7$ and another one is $35$. Because the quadrilateral is inscribed in a circle, the sum of opposite angles equals 180 degrees.

1. Problem Description

The problem asks to solve for $x$ given the expressions for the measures of the angles in a quadrilateral inscribed in a circle, where two angles are $3x+11$ and $6x-7$ and another one is $35$. Because the quadrilateral is inscribed in a circle, the sum of opposite angles equals 180 degrees.

2. Solution Steps

Let the four angles of the quadrilateral be $A, B, C,$ and $D$. We are given that $A = 3x+11$, $B = 6x-7$, and $C = 35$. Since the sum of angles in a quadrilateral is 360 degrees, $A+B+C+D = 360$. Also, opposite angles of a cyclic quadrilateral are supplementary, i.e., add up to 180 degrees.

We have two cases:

Case 1: $A$ and $C$ are opposite angles. Then $A+C = 180$, so $3x+11+35 = 180$, i.e., $3x+46=180$.

Case 2: $B$ and $C$ are opposite angles. Then $B+C = 180$, so $6x-7+35 = 180$, i.e., $6x+28 = 180$.

In Case 1, $3x+46 = 180$.

$3x = 180 - 46 = 134$.

$x = \frac{134}{3}$.

In Case 2, $6x+28 = 180$.

$6x = 180 - 28 = 152$.

$x = \frac{152}{6} = \frac{76}{3}$.

Since the problem has right angle symbols where they meet at the center of the circle, this implies that the angles $3x+11$ and $6x-7$ are opposite angles. Thus,

$3x+11 + 6x-7 = 180$.

$9x + 4 = 180$.

$9x = 176$.

$x = \frac{176}{9}$.

3. Final Answer

176/9