Rational Inequality 21Dbe6

1. **State the problem:** Solve the inequality $$\frac{-x + 5}{x - 7} \geq 0$$ and express the solution set in interval notation. 2. **Identify critical points:** The numerator is zero when $$-x + 5 = 0 \Rightarrow x = 5$$. The denominator is zero when $$x - 7 = 0 \Rightarrow x = 7$$ (excluded from domain). 3. **Determine intervals:** The critical points divide the real line into three intervals: - $$(-\infty, 5)$$ - $$(5, 7)$$ - $$(7, \infty)$$ 4. **Test each interval:** - For $$x < 5$$, pick $$x = 0$$: $$\frac{-0 + 5}{0 - 7} = \frac{5}{-7} = -\frac{5}{7} < 0$$ (does not satisfy inequality). - For $$5 < x < 7$$, pick $$x = 6$$: $$\frac{-6 + 5}{6 - 7} = \frac{-1}{-1} = 1 > 0$$ (satisfies inequality). - For $$x > 7$$, pick $$x = 8$$: $$\frac{-8 + 5}{8 - 7} = \frac{-3}{1} = -3 < 0$$ (does not satisfy inequality). 5. **Check equality at critical points:** - At $$x = 5$$, numerator zero, denominator nonzero, so $$\frac{0}{5 - 7} = 0$$ satisfies $$\geq 0$$. - At $$x = 7$$, denominator zero, undefined, exclude from solution. 6. **Write solution set:** $$[5, 7)$$ **Final answer:** The solution set is $$[5, 7)$$.