
9.9E: Exercises


Practice Makes Perfect

Exercise $$\PageIndex{11}$$ Solve Quadratic Inequalities Graphically

In the following exercises,

1. Solve graphically
2. Write the solution in interval notation
1. $$x^{2}+6 x+5>0$$
2. $$x^{2}+4 x-12<0$$
3. $$x^{2}+4 x+3 \leq 0$$
4. $$x^{2}-6 x+8 \geq 0$$
5. $$-x^{2}-3 x+18 \leq 0$$
6. $$-x^{2}+2 x+24<0$$
7. $$-x^{2}+x+12 \geq 0$$
8. $$-x^{2}+2 x+15>0$$

1.

1. Figure 9.8.16
2. $$(-\infty,-5) \cup(-1, \infty)$$

3.

1. Figure 9.8.17
2. $$[-3,-1]$$

5.

1. Figure 9.8.18
2. $$(-\infty,-6] \cup[3, \infty)$$

7.

1. Figure 9.8.19
2. $$[-3,4]$$
Exercise $$\PageIndex{12}$$ Solve Quadratic Inequalities Graphically

In the following exercises, solve each inequality algebraically and write any solution in interval notation.

1. $$x^{2}+3 x-4 \geq 0$$
2. $$x^{2}+x-6 \leq 0$$
3. $$x^{2}-7 x+10<0$$
4. $$x^{2}-4 x+3>0$$
5. $$x^{2}+8 x>-15$$
6. $$x^{2}+8 x<-12$$
7. $$x^{2}-4 x+2 \leq 0$$
8. $$-x^{2}+8 x-11<0$$
9. $$x^{2}-10 x>-19$$
10. $$x^{2}+6 x<-3$$
11. $$-6 x^{2}+19 x-10 \geq 0$$
12. $$-3 x^{2}-4 x+4 \leq 0$$
13. $$-2 x^{2}+7 x+4 \geq 0$$
14. $$2 x^{2}+5 x-12>0$$
15. $$x^{2}+3 x+5>0$$
16. $$x^{2}-3 x+6 \leq 0$$
17. $$-x^{2}+x-7>0$$
18. $$-x^{2}-4 x-5<0$$
19. $$-2 x^{2}+8 x-10<0$$
20. $$-x^{2}+2 x-7 \geq 0$$

1. $$(-\infty,-4] \cup[1, \infty)$$

3. $$(2,5)$$

5. $$(-\infty,-5) \cup(-3, \infty)$$

7. $$[2-\sqrt{2}, 2+\sqrt{2}]$$

9. $$(-\infty, 5-\sqrt{6}) \cup(5+\sqrt{6}, \infty)$$

11. $$\left(-\infty,-\frac{5}{2}\right] \cup\left[-\frac{2}{3}, \infty\right)$$

13. $$\left[-\frac{1}{2}, 4\right]$$

15. $$(-\infty, \infty)$$

17. no solution

19. $$(-\infty, \infty)$$

Exercise $$\PageIndex{13}$$ Writing Exercises
1. Explain critical points and how they are used to solve quadratic inequalities algebraically.
2. Solve $$x^{2}+2x≥8$$ both graphically and algebraically. Which method do you prefer, and why?
3. Describe the steps needed to solve a quadratic inequality graphically.
4. Describe the steps needed to solve a quadratic inequality algebraically.