Rational Inequality Ac67F9

1. **State the problem:** Solve the inequality $$\frac{x - 2}{x - 1} \leq 0$$ and express the solution set in interval notation. 2. **Rewrite the inequality:** Move all terms to the left side and define $$f(x) = \frac{x - 2}{x - 1}$$ 3. **Find zeros and undefined points:** - Numerator zero: $$x - 2 = 0 \Rightarrow x = 2$$ - Denominator zero (undefined): $$x - 1 = 0 \Rightarrow x = 1$$ 4. **Determine intervals based on critical points:** The critical points split the real line into three intervals: - $$(-\infty, 1)$$ - $$(1, 2)$$ - $$(2, \infty)$$ 5. **Test the sign of $$f(x)$$ in each interval:** - For $$x < 1$$, pick $$x=0$$: $$f(0) = \frac{0 - 2}{0 - 1} = \frac{-2}{-1} = 2 > 0$$ - For $$1 < x < 2$$, pick $$x=1.5$$: $$f(1.5) = \frac{1.5 - 2}{1.5 - 1} = \frac{-0.5}{0.5} = -1 < 0$$ - For $$x > 2$$, pick $$x=3$$: $$f(3) = \frac{3 - 2}{3 - 1} = \frac{1}{2} = 0.5 > 0$$ 6. **Include points where $$f(x) = 0$$:** At $$x=2$$, $$f(2) = 0$$, so include $$x=2$$ in the solution. 7. **Exclude points where $$f(x)$$ is undefined:** At $$x=1$$, $$f(x)$$ is undefined, so exclude $$x=1$$. 8. **Write the solution set:** The inequality $$\frac{x - 2}{x - 1} \leq 0$$ holds where $$f(x) \leq 0$$, which is on the interval $$(1, 2]$$. 9. **Summary table:** | Interval | Sign of $$f(x)$$ | |----------------|-----------------| | $$(-\infty, 1)$$ | Positive (+) | | $$(1, 2)$$ | Negative (-) | | $$(2, \infty)$$ | Positive (+) | 10. **Final answer:** $$\boxed{(1, 2]}$$