1. **State the problem:** We have two secants intersecting outside a circle, forming an angle of $4x^\circ$. The intercepted arcs inside the circle are $35^\circ$ and $(9x + 26)^\circ$. We need to find the value of $x$. 2. **Formula used:** The angle formed by two secants intersecting outside a circle is half the difference of the measures of the intercepted arcs: $$\text{Angle} = \frac{1}{2} |\text{arc}_1 - \text{arc}_2|$$ 3. **Set up the equation:** $$4x = \frac{1}{2} |(9x + 26) - 35|$$ Simplify inside the absolute value: $$4x = \frac{1}{2} |9x + 26 - 35| = \frac{1}{2} |9x - 9|$$ 4. **Multiply both sides by 2 to eliminate the fraction:** $$2 \times 4x = 2 \times \frac{1}{2} |9x - 9|$$ $$8x = |9x - 9|$$ 5. **Solve the absolute value equation:** Case 1: $9x - 9 = 8x$ $$9x - 9 = 8x$$ $$9x - 8x = 9$$ $$x = 9$$ Case 2: $9x - 9 = -8x$ $$9x - 9 = -8x$$ $$9x + 8x = 9$$ $$17x = 9$$ $$x = \frac{9}{17}$$ 6. **Check for validity:** For $x=9$, the arcs are: $$9x + 26 = 9(9) + 26 = 81 + 26 = 107^\circ$$ The other arc is $35^\circ$. The difference is $|107 - 35| = 72^\circ$. Angle should be half of 72, which is 36. Check if $4x = 36$: $$4 \times 9 = 36$$ Valid. For $x=\frac{9}{17} \approx 0.529$, arcs: $$9x + 26 = 9 \times 0.529 + 26 \approx 4.76 + 26 = 30.76^\circ$$ Difference: $$|30.76 - 35| = 4.24^\circ$$ Half difference: $$\frac{4.24}{2} = 2.12^\circ$$ Angle $4x$: $$4 \times 0.529 = 2.12^\circ$$ Also valid. 7. **Conclusion:** Both $x=9$ and $x=\frac{9}{17}$ satisfy the equation. Depending on the context (usually arcs are larger than 35°), $x=9$ is the more reasonable solution. **Final answer:** $$x = 9$$