1. **Problem:** Find the Cartesian equation for the parametric curve given by $$x = \frac{t}{t - 1}, \quad y = \frac{t - 2}{t + 1}, \quad -1 < t < 1.$$\n\n2. **Goal:** Eliminate the parameter $t$ to express $y$ solely in terms of $x$.\n\n3. **Step 1:** Start with the expression for $x$: $$x = \frac{t}{t - 1}.$$ Multiply both sides by $(t - 1)$ to get: $$x(t - 1) = t.$$\n\n4. **Step 2:** Distribute $x$: $$xt - x = t.$$ Rearrange to isolate terms with $t$: $$xt - t = x.$$ Factor $t$ out: $$t(x - 1) = x.$$\n\n5. **Step 3:** Solve for $t$: $$t = \frac{x}{x - 1}.$$\n\n6. **Step 4:** Substitute $t$ into the expression for $y$: $$y = \frac{t - 2}{t + 1} = \frac{\frac{x}{x - 1} - 2}{\frac{x}{x - 1} + 1}.$$\n\n7. **Step 5:** Simplify numerator: $$\frac{x}{x - 1} - 2 = \frac{x - 2(x - 1)}{x - 1} = \frac{x - 2x + 2}{x - 1} = \frac{2 - x}{x - 1}.$$\n\n8. **Step 6:** Simplify denominator: $$\frac{x}{x - 1} + 1 = \frac{x + (x - 1)}{x - 1} = \frac{2x - 1}{x - 1}.$$\n\n9. **Step 7:** Now, $$y = \frac{\frac{2 - x}{x - 1}}{\frac{2x - 1}{x - 1}} = \frac{2 - x}{2x - 1}.$$\n\n10. **Final Cartesian equation:** $$\boxed{y = \frac{2 - x}{2x - 1}}.$$\n\nThis equation relates $x$ and $y$ without the parameter $t$.\n\n**Note:** The domain restrictions on $t$ translate to restrictions on $x$ and $y$ accordingly, but the Cartesian equation is valid for the corresponding $x$ values where denominators are not zero.