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Homework: Inequalities

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

Summary: This module provides practice problems related to inequalities.

Exercise 1

2x + 7 ≤ 4x + 4 2x + 7 ≤ 4x + 4 size 12{2x+7 <= 4x+4} {}

  • a. Solve for xx size 12{x} {}.
  • b. Draw a number line, and show where the solution set to this problem is.
  • c. Pick an xx size 12{x} {}-value which, according to your drawing, is inside the solution set. Plug it into the original inequality 2x+7≤4x+42x+7≤4x+4 size 12{2x+7 <= 4x+4} {}. Does the inequality hold true?
  • d. Pick an xx size 12{x} {}-value which, according to your drawing, is outside the solution set. Plug it into the original inequality 2x+7≤4x+42x+7≤4x+4 size 12{2x+7 <= 4x+4} {}. Does the inequality hold true?

Exercise 2

14 − 2x < 20 14 − 2x < 20 size 12{"14" - 2x<"20"} {}

  • a. Solve for xx size 12{x} {}.
  • b. Draw a number line, and show where the solution set to this problem is.
  • c. Pick an xx size 12{x} {}-value which, according to your drawing, is inside the solution set. Plug it into the original inequality 14−2x<2014−2x<20 size 12{"14" - 2x<"20"} {}. Does the inequality hold true?
  • d. Pick an xx size 12{x} {}-value which, according to your drawing, is outside the solution set. Plug it into the original inequality 14−2x<2014−2x<20 size 12{"14" - 2x<"20"} {}. Does the inequality hold true?

Exercise 3

− 10 < 3x + 2 ≤ 5 − 10 < 3x + 2 ≤ 5 size 12{ - "10"<3x+2 <= 5} {}

  • a. Solve for xx size 12{x} {}.
  • b. Draw a number line, and show where the solution set to this problem is.
  • c. Pick an xx size 12{x} {}-value which, according to your drawing, is inside the solution set. Plug it into the original inequality −10<3x+2≤5−10<3x+2≤5 size 12{ - "10"<3x+2 <= 5} {}. Does the inequality hold true?
  • d. Pick an xx size 12{x} {}-value which, according to your drawing, is outside the solution set. Plug it into the original inequality −10<3x+2≤5−10<3x+2≤5 size 12{ - "10"<3x+2 <= 5} {}. Does the inequality hold true?

Exercise 4

x<3x<3 size 12{x<3} {} and x<7x<7 size 12{x<7} {}. Draw a number line, and show where the solution set to this problem is.

Exercise 5

x<3x<3 size 12{x<3} {} or x<7x<7 size 12{x<7} {}. Draw a number line, and show where the solution set to this problem is.

Exercise 6

x − 2y ≥ 4 x − 2y ≥ 4 size 12{x - 2y >= 4} {}

  • a. Solve for yy size 12{y} {}.
  • b. Now—for the moment—let’s pretend that your equation said equals instead of “greater than” or “less than.” Then it would be the equation for a line. Find the slope and the y-intercept of that line, and graph it.
  • Slope: _________
  • y-intercept__________
  • c. Now, pick any point (x,y)(x,y) size 12{ \( x,y \) } {} that is above that line. Plug the xx size 12{x} {} and yy size 12{y} {} coordinates into your inequality from part (a). Does this point fit the inequality? (Show your work…)
  • d. Now, pick any point (x,y)(x,y) size 12{ \( x,y \) } {} that is below that line. Plug the xx size 12{x} {} and yy size 12{y} {} coordinates into your inequality from part (a). Does this point fit the inequality? (Show your work…)
  • e. So, is the solution to the inequality the points below or above the line? Shade the appropriate region on your graph.

Exercise 7

Using a similar technique, draw the graph of y≥x2y≥x2 size 12{y >= x rSup { size 8{2} } } {}. (If you don’t remember what the graph of y≥x2y≥x2 size 12{y >= x rSup { size 8{2} } } {} looks like, try plotting a few points!)

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