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Absolute Value Inequalities

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

Summary: This module provides sample problems which develop concepts related to absolute value inequalities.

Exercise 1

∣ x ∣ ≤ 7 ∣ x ∣ ≤ 7 size 12{ lline x rline <= 7} {}

  • a. Solve.
  • b. Graph your solution on a number line
  • c. Choose a point that is in your solution set, and test it in the original inequality. Does it work?
  • d. Choose a point that is not in your solution set, and test it in the original inequality. Does it work?

Exercise 2

∣ 2x + 3 ∣ ≤ 7 ∣ 2x + 3 ∣ ≤ 7 size 12{ lline 2x+3 rline <= 7} {}

  • a. Write down the solution for what 2x+32x+3 size 12{2x+3} {} has to be. This should look exactly like your answer to number 1, except with a (2x+3)(2x+3) size 12{ \( 2x+3 \) } {} instead of an (x)(x) size 12{ \( x \) } {}.
  • b. Now, solve that inequality for xx size 12{x} {}.
  • c. Graph your solution on a number line
  • d. Choose a point that is in your solution set, and test it in the original inequality. Does it work?
  • e. Choose a point that is not in your solution set, and test it in the original inequality. Does it work?

Exercise 3

4 ∣ 3x − 6 ∣ + 7 > 19 4 ∣ 3x − 6 ∣ + 7 > 19 size 12{4 lline 3x - 6 rline +7>"19"} {}

  • a. Solve for ∣3x−6∣∣3x−6∣ size 12{ lline 3x - 6 rline } {}. (That is, leave the ∣3x−6∣∣3x−6∣ size 12{ lline 3x - 6 rline } {} part alone, but get rid of all the stuff around it.)
  • b. Write down the inequality for what (3x−6)(3x−6) size 12{ \( 3x - 6 \) } {} has to be.
  • c. Now, solve that inequality for xx size 12{x} {}.
  • d. Graph your solution on a number line
  • e. Choose a point that is in your solution set, and test it in the original inequality. Does it work?
  • f. Choose a point that is not in your solution set, and test it in the original inequality. Does it work?

Exercise 4

∣ 3x − 4 ∣ 2 + 6 < 3 ∣ 3x − 4 ∣ 2 + 6 < 3 size 12{ { { lline 3x - 4 rline } over {2} } +6<3} {}

  • a. Solve for xx size 12{x} {}. (You know the drill by now!)
  • b. Graph your solution on a number line
  • c. Choose a point that is in your solution set, and test it in the original inequality. Does it work?
  • d. Choose a point that is not in your solution set, and test it in the original inequality. Does it work?

Exercise 5

6 ∣ 2x 2 − 17 x − 85 ∣ + 5 ≥ 3 6 ∣ 2x 2 − 17 x − 85 ∣ + 5 ≥ 3 size 12{6 lline 2x rSup { size 8{2} } - "17"x - "85" rline +5 >= 3} {}

  • a. Solve for xx size 12{x} {}.
  • b. Graph your solution on a number line
  • c. Choose a point that is in your solution set, and test it in the original inequality. Does it work?
  • d. Choose a point that is not in your solution set, and test it in the original inequality. Does it work?

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