19.5E: Exercises
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Practice Makes Perfect
Exercise \(\PageIndex{11}\) Solve Quadratic Inequalities Graphically
In the following exercises,
- Solve graphically
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Write the solution in interval notation
- \(x^{2}+6 x+5>0\)
- \(x^{2}+4 x-12<0\)
- \(x^{2}+4 x+3 \leq 0\)
- \(x^{2}-6 x+8 \geq 0\)
- \(-x^{2}-3 x+18 \leq 0\)
- \(-x^{2}+2 x+24<0\)
- \(-x^{2}+x+12 \geq 0\)
- \(-x^{2}+2 x+15>0\)
- Answer
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1.
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Figure 9.8.16 - \((-\infty,-5) \cup(-1, \infty)\)
3.
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Figure 9.8.17 - \([-3,-1]\)
5.
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Figure 9.8.18 - \((-\infty,-6] \cup[3, \infty)\)
7.
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Figure 9.8.19 - \([-3,4]\)
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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
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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
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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.
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?