Quadratic Inequalities 98C7F1

1. **Problem 1:** Solve the inequality $$x^2 - 4x \leq 0$$ 2. **Step 1:** Factor the quadratic expression. $$x^2 - 4x = x(x - 4)$$ 3. **Step 2:** Set the product less than or equal to zero. $$x(x - 4) \leq 0$$ 4. **Step 3:** Find the critical points by setting each factor equal to zero. $$x = 0 \quad \text{or} \quad x - 4 = 0 \Rightarrow x = 4$$ 5. **Step 4:** Test intervals determined by critical points: $(-\infty, 0)$, $(0, 4)$, and $(4, \infty)$. - For $x < 0$, choose $x = -1$: $(-1)(-1 - 4) = (-1)(-5) = 5 > 0$ (False) - For $0 < x < 4$, choose $x = 2$: $(2)(2 - 4) = 2(-2) = -4 \leq 0$ (True) - For $x > 4$, choose $x = 5$: $(5)(5 - 4) = 5(1) = 5 > 0$ (False) 6. **Step 5:** Include points where expression equals zero: $x=0$ and $x=4$. 7. **Answer for Problem 1:** $$0 \leq x \leq 4$$ --- 8. **Problem 2:** Solve the inequality $$x^2 - 16 > 0$$ 9. **Step 1:** Factor the difference of squares. $$x^2 - 16 = (x - 4)(x + 4)$$ 10. **Step 2:** Set the product greater than zero. $$(x - 4)(x + 4) > 0$$ 11. **Step 3:** Find critical points. $$x - 4 = 0 \Rightarrow x = 4$$ $$x + 4 = 0 \Rightarrow x = -4$$ 12. **Step 4:** Test intervals: $(-\infty, -4)$, $(-4, 4)$, $(4, \infty)$. - For $x < -4$, choose $x = -5$: $(-5 - 4)(-5 + 4) = (-9)(-1) = 9 > 0$ (True) - For $-4 < x < 4$, choose $x = 0$: $(0 - 4)(0 + 4) = (-4)(4) = -16 < 0$ (False) - For $x > 4$, choose $x = 5$: $(5 - 4)(5 + 4) = (1)(9) = 9 > 0$ (True) 13. **Step 5:** Since inequality is strict ($>$), exclude $x = -4$ and $x = 4$. 14. **Answer for Problem 2:** $$x < -4 \quad \text{or} \quad x > 4$$