Question: The solution set of the quadratic inequality $x^2 - 5x + 4 \leq 0$ is ________.
1. **State the problem:** We need to find the solution set for the quadratic inequality $$x^2 - 5x + 4 \leq 0.$$
2. **Recall the formula and rules:** To solve quadratic inequalities, first find the roots of the corresponding quadratic equation $$x^2 - 5x + 4 = 0.$$ These roots divide the number line into intervals. We test each interval to determine where the inequality holds.
3. **Find the roots:** Solve $$x^2 - 5x + 4 = 0$$ using factoring:
$$x^2 - 5x + 4 = (x - 1)(x - 4) = 0.$$
So, the roots are $$x = 1$$ and $$x = 4.$$
4. **Determine intervals:** The roots split the number line into three intervals:
- $$(-\infty, 1)$$
- $$[1, 4]$$
- $$ (4, \infty)$$
5. **Test each interval:**
- For $$x < 1$$, pick $$x = 0$$:
$$0^2 - 5(0) + 4 = 4 > 0,$$ so inequality $$\leq 0$$ is false here.
- For $$1 \leq x \leq 4$$, pick $$x = 2$$:
$$2^2 - 5(2) + 4 = 4 - 10 + 4 = -2 \leq 0,$$ true.
- For $$x > 4$$, pick $$x = 5$$:
$$5^2 - 5(5) + 4 = 25 - 25 + 4 = 4 > 0,$$ false.
6. **Conclusion:** The solution set is $$1 \leq x \leq 4.$$
7. **Answer choice:** This corresponds to option b.) $$1 \leq x \leq 4.$$