1. The problem involves finding the value of $x$ given angles in a circle, where angles such as $x^\circ$, $2x^\circ$, $3x^\circ$, $-x^\circ$, and $130^\circ$ appear in various configurations. 2. Key circle theorems to use: - The sum of angles around a point is $360^\circ$. - Angles in a triangle sum to $180^\circ$. - Angles subtended by the same arc are equal. - The angle between a tangent and a chord is equal to the angle in the alternate segment. 3. From the given angles, identify relationships. For example, if angles $x^\circ$, $2x^\circ$, and $3x^\circ$ are parts of a triangle or around a point, their sum must satisfy the relevant theorem. 4. Suppose the angles $x^\circ$, $2x^\circ$, and $3x^\circ$ form a triangle, then: $$x + 2x + 3x = 180$$ $$6x = 180$$ $$x = 30$$ 5. Check consistency with other given angles such as $130^\circ$ and $-x^\circ$ (which means an angle of $-30^\circ$ if $x=30$), ensuring all angle sums and circle theorems hold. 6. Therefore, the value of $x$ is $30^\circ$. Final answer: $x = 30^\circ$