1. **State the problem:** We are given a point $P = \left(x, \frac{2}{3}\right)$ that lies on the unit circle centered at the origin. We need to find the value of $x$ in simplest form. 2. **Recall the unit circle equation:** The unit circle has radius 1 and is centered at the origin, so any point $(x,y)$ on it satisfies: $$x^2 + y^2 = 1$$ 3. **Substitute the given $y$ value:** Here, $y = \frac{2}{3}$, so: $$x^2 + \left(\frac{2}{3}\right)^2 = 1$$ 4. **Simplify the equation:** $$x^2 + \frac{4}{9} = 1$$ 5. **Isolate $x^2$:** $$x^2 = 1 - \frac{4}{9} = \frac{9}{9} - \frac{4}{9} = \frac{5}{9}$$ 6. **Take the square root:** $$x = \pm \sqrt{\frac{5}{9}} = \pm \frac{\sqrt{5}}{3}$$ 7. **Determine the sign of $x$:** The point $P$ is in the upper-left quadrant, where $x$ is negative and $y$ is positive. Since $y=\frac{2}{3}>0$, $x$ must be negative. **Final answer:** $$x = -\frac{\sqrt{5}}{3}$$