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Homework: Quadratic Inequalities

Module by: Kenny M. Felder. E-mail the author

Summary: Some homework problems on quadratic inequalities.

Exercise 1

x 2 + 8x + 7 > 2x + 3 x 2 + 8x + 7 > 2x + 3 size 12{x rSup { size 8{2} } +8x+7>2x+3} {}

  • a. Collect all the terms on one side, so the other side of the inequality is zero. Then graph the function by finding the zeros.
  • b. Based on your graph, for what xx size 12{x} {}-values is this inequality true?
  • c. Based on your answer, choose one xx size 12{x} {}-value for which the inequality should hold, and one for which it should not. Check to make sure they both do what they should.

Exercise 2

2x 2 + 8x + 8 ≤ 0 2x 2 + 8x + 8 ≤ 0 size 12{2x rSup { size 8{2} } +8x+8 <= 0} {}

  • a. Graph
  • b. Based on your graph, for what xx size 12{x} {}-values is this inequality true?
  • c. Based on your answer, choose one xx size 12{x} {}-value for which the inequality should hold, and one for which it should not. Check to make sure they both do what they should.

Exercise 3

− 2x 2 + 8x > 9 − 2x 2 + 8x > 9 size 12{ - 2x rSup { size 8{2} } +8x>9} {}

  • a. Collect terms and graph
  • b. Based on your graph, for what xx size 12{x} {}-values is this inequality true?
  • c. Based on your answer, choose one xx size 12{x} {}-value for which the inequality should hold, and one for which it should not. Check to make sure they both do what they should.

Exercise 4

− x 2 + 4x + 3 > 0 − x 2 + 4x + 3 > 0 size 12{ - x rSup { size 8{2} } +4x+3>0} {}

  • a. Graph the function
  • b. Based on your graph, for what xx size 12{x} {}-values is this inequality true?
  • c. Based on your answer, choose one xx size 12{x} {}-value for which the inequality should hold, and one for which it should not. Check to make sure they both do what they should.

Exercise 5

x 2 > x x 2 > x size 12{x rSup { size 8{2} } >x} {}

  • a. Collect terms and graph
  • b. Based on your graph, for what xx size 12{x} {}-values is this inequality true?
  • c. Now, let’s solve the original equation a different way—divide both sides by xx size 12{x} {}. Did you get the same answer this way? If not, which one is correct? (Answer by trying points.) What went wrong with the other one?

Exercise 6

x 2 + 6x + c < 0 x 2 + 6x + c < 0 size 12{x rSup { size 8{2} } +6x+c<0} {}

  • a. For what values of cc size 12{c} {} will this inequality be true in some range?
  • b. For what values of cc size 12{c} {} will this inequality never be true?
  • c. For what values of cc size 12{c} {} will this inequality always be true?

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