Question: Solve the rational inequality and graph the solution set on a real number line. Express the solution set in interval notation.
$$\frac{x - 5}{x - 4} < 0$$
Solve the inequality. What is the solution set? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The solution set is ☐.
(Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.)
B. The solution set is the empty set.
1. **State the problem:** Solve the inequality $$\frac{x - 5}{x - 4} < 0$$ and express the solution set in interval notation.
2. **Identify critical points:** The numerator $$x - 5 = 0$$ when $$x = 5$$, and the denominator $$x - 4 = 0$$ when $$x = 4$$.
3. **Determine intervals:** The critical points divide the real number line into three intervals:
- $$(-\infty, 4)$$
- $$(4, 5)$$
- $$(5, \infty)$$
4. **Test each interval:** Choose a test point in each interval to determine the sign of $$\frac{x - 5}{x - 4}$$.
- For $$x = 3$$ in $$(-\infty, 4)$$:
$$\frac{3 - 5}{3 - 4} = \frac{-2}{-1} = 2 > 0$$ (positive)
- For $$x = 4.5$$ in $$(4, 5)$$:
$$\frac{4.5 - 5}{4.5 - 4} = \frac{-0.5}{0.5} = -1 < 0$$ (negative)
- For $$x = 6$$ in $$(5, \infty)$$:
$$\frac{6 - 5}{6 - 4} = \frac{1}{2} = 0.5 > 0$$ (positive)
5. **Determine where the inequality holds:** We want $$\frac{x - 5}{x - 4} < 0$$, so the solution is where the expression is negative, which is on $$(4, 5)$$.
6. **Check endpoints:**
- At $$x = 4$$, the denominator is zero, so $$x = 4$$ is excluded.
- At $$x = 5$$, the numerator is zero, so $$\frac{0}{5 - 4} = 0$$, which does not satisfy the strict inequality $$< 0$$, so $$x = 5$$ is excluded.
7. **Final solution set:** $$\boxed{(4, 5)}$$
**Answer:** The solution set is $$(4, 5)$$.
**Graph:** The solution set is the open interval between 4 and 5 on the real number line, excluding the endpoints.