1. **State the problem:** Solve the inequality $$\frac{(x-2)(x+1)}{(x-1)} \leq 0$$ and find the sum of all positive integers satisfying it. 2. **Identify critical points:** The numerator is zero at $$x=2$$ and $$x=-1$$. The denominator is zero at $$x=1$$ (undefined point). 3. **Determine intervals:** The critical points divide the number line into intervals: $$(-\infty, -1), (-1, 1), (1, 2), (2, \infty)$$ 4. **Test signs on each interval:** - For $$x < -1$$, pick $$x=-2$$: numerator is $$( -2-2)(-2+1) = (-4)(-1) = 4 > 0$$, denominator $$( -2-1) = -3 < 0$$, so fraction is $$\frac{+}{-} = -$$ (negative). - For $$-1 < x < 1$$, pick $$x=0$$: numerator $$(0-2)(0+1) = (-2)(1) = -2 < 0$$, denominator $$(0-1) = -1 < 0$$, fraction is $$\frac{-}{-} = +$$ (positive). - For $$1 < x < 2$$, pick $$x=1.5$$: numerator $$(1.5-2)(1.5+1) = (-0.5)(2.5) = -1.25 < 0$$, denominator $$(1.5-1) = 0.5 > 0$$, fraction is $$\frac{-}{+} = -$$ (negative). - For $$x > 2$$, pick $$x=3$$: numerator $$(3-2)(3+1) = (1)(4) = 4 > 0$$, denominator $$(3-1) = 2 > 0$$, fraction is $$\frac{+}{+} = +$$ (positive). 5. **Include points where fraction equals zero:** The fraction equals zero when numerator is zero and denominator is not zero, so at $$x=-1$$ and $$x=2$$. 6. **Exclude points where denominator is zero:** At $$x=1$$, fraction is undefined. 7. **Write solution set:** Inequality $$\leq 0$$ means fraction is negative or zero. From sign analysis: - Negative intervals: $$(-\infty, -1)$$ and $$(1, 2)$$ - Zero points: $$x=-1$$ and $$x=2$$ So solution is $$(-\infty, -1] \cup (1, 2]$$. 8. **Find positive integers in solution:** Positive integers are $$1, 2, 3, ...$$ From solution, positive integers must satisfy $$1 < x \leq 2$$. Only integer is $$2$$. 9. **Sum of all positive integers satisfying inequality:** $$2$$. **Final answer:** $$2$$