1. **State the problem:** We are given a circle with two chords intersecting at point L inside the circle. The angles and arcs are labeled as follows: - Angle at L between rays LJ and LI is 85.5° - Arc HJ measures (15x + 24)° - Arc IK measures (42x - 6)° We need to find the value of $x$. 2. **Formula used:** When two chords intersect inside a circle, the measure of the angle formed is half the sum of the measures of the intercepted arcs. Mathematically, $$\text{Angle} = \frac{1}{2} (\text{Arc}_1 + \text{Arc}_2)$$ 3. **Apply the formula:** Here, $$85.5 = \frac{1}{2} \big((15x + 24) + (42x - 6)\big)$$ 4. **Simplify inside the parentheses:** $$(15x + 24) + (42x - 6) = 15x + 42x + 24 - 6 = 57x + 18$$ So, $$85.5 = \frac{1}{2} (57x + 18)$$ 5. **Multiply both sides by 2 to eliminate the fraction:** $$2 \times 85.5 = 57x + 18$$ $$171 = 57x + 18$$ 6. **Isolate $x$ by subtracting 18 from both sides:** $$171 - 18 = 57x$$ $$153 = 57x$$ 7. **Divide both sides by 57 to solve for $x$:** $$x = \frac{153}{57}$$ Show cancellation: $$x = \frac{\cancel{153}^{3 \times 51}}{\cancel{57}^{3 \times 19}} = \frac{51}{19}$$ 8. **Calculate the decimal value:** $$x \approx 2.6842$$ **Final answer:** $$x = \frac{51}{19} \approx 2.68$$