4.8.1E: Exercises

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Feb 8, 2022
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- 4.8.1: Solve Quadratic Inequalities
- 4.8.2: Solve Rational Inequalities

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Practice Makes Perfect

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

In the following exercises,

Solve graphically
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. $-x^{2}+2 x+15>0$

Answer

$The graph shown is an upward-facing parabola with vertex (negative 3, negative 4) and y-intercept (0,5).$
Figure 9.8.16
$(-\infty,-5) \cup(-1, \infty)$

$The graph shown is an upward facing parabola with vertex (negative 2, negative 1) and y-intercept (0,3).$
Figure 9.8.17
$[-3,-1]$

$The graph shown is a downward-facing parabola with vertex (negative 1 and 5 tenths, 20) and y-intercept (0, 18).$
Figure 9.8.18
$(-\infty,-6] \cup[3, \infty)$

$The graph shown is a downward facing parabola with a y-intercept of (0, 12) and x-intercepts (negative 3, 0) and (4, 0).$
Figure 9.8.19
$[-3,4]$

Exercise $\PageIndex{12}$ Solve Quadratic Inequalities Graphically

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

$x^{2}+3 x-4 \geq 0$
$x^{2}+x-6 \leq 0$
$x^{2}-7 x+10<0$
$x^{2}-4 x+3>0$
$x^{2}+8 x>-15$
$x^{2}+8 x<-12$
$x^{2}-4 x+2 \leq 0$
$-x^{2}+8 x-11<0$
$x^{2}-10 x>-19$
$x^{2}+6 x<-3$
$-6 x^{2}+19 x-10 \geq 0$
$-3 x^{2}-4 x+4 \leq 0$
$-2 x^{2}+7 x+4 \geq 0$
$2 x^{2}+5 x-12>0$
$x^{2}+3 x+5>0$
$x^{2}-3 x+6 \leq 0$
$-x^{2}+x-7>0$
$-x^{2}-4 x-5<0$
$-2 x^{2}+8 x-10<0$
$-x^{2}+2 x-7 \geq 0$

Answer

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

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

Answer

1. Answers may vary.

3. Answers may vary.

Self Check

a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This figure is a list to assess your understanding of the concepts presented in this section. It has 4 columns labeled I canâ€¦, Confidently, With some help, and No-I donâ€™t get it! Below I canâ€¦, there is solve quadratic inequalities graphically and solve quadratic inequalities algebraically. The other columns are left blank for you to check you understanding. — Figure 9.8.20

b. On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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Practice Makes Perfect

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

Exercise $\PageIndex{12}$ Solve Quadratic Inequalities Graphically

Exercise $\PageIndex{13}$ Writing Exercises

Self Check

Practice Makes Perfect

Exercise 4.8.1E.11\PageIndex{11} Solve Quadratic Inequalities Graphically

Exercise 4.8.1E.12\PageIndex{12} Solve Quadratic Inequalities Graphically

Exercise 4.8.1E.13\PageIndex{13} Writing Exercises

Self Check

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

Exercise $\PageIndex{12}$ Solve Quadratic Inequalities Graphically

Exercise $\PageIndex{13}$ Writing Exercises