1. **State the problem:** We are given two angles at point Y, one measuring $(4x - 1)^\circ$ and the other $(2x + 5)^\circ$, and these angles are adjacent and form a right angle (90°). 2. **Formula and rule:** Adjacent angles that form a right angle sum to 90°. So, we use the equation: $$ (4x - 1) + (2x + 5) = 90 $$ 3. **Solve the equation:** Combine like terms: $$ 4x - 1 + 2x + 5 = 90 $$ $$ (4x + 2x) + (-1 + 5) = 90 $$ $$ 6x + 4 = 90 $$ 4. **Isolate $x$:** $$ 6x + 4 = 90 $$ Subtract 4 from both sides: $$ 6x + \cancel{4} - \cancel{4} = 90 - 4 $$ $$ 6x = 86 $$ 5. **Divide both sides by 6:** $$ \frac{6x}{6} = \frac{86}{6} $$ $$ x = \frac{86}{6} $$ Simplify the fraction: $$ x = \frac{43}{3} \approx 14.33 $$ 6. **Final answer:** $$ x = \frac{43}{3} $$ or approximately 14.33 degrees. This means the value of $x$ that makes the two angles sum to a right angle is $\frac{43}{3}$.