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Inequalities

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

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

Exercise 1

4<64<6 size 12{4<6} {} (I think we can all agree on that, yes?)

  • a. Add 4 to both sides of the equation. ___________ Is it still true?
  • b. Add −4−4 size 12{ - 4} {} to both sides of the (original) equation. ___________ Is it still true?
  • c. Subtract 10 from both sides of the (original) equation. ___________ Is it still true?
  • d. Multiply both sides of the (original) equation by 4. ___________ Is it still true?
  • e. Divide both sides of the (original) equation by 2. ___________ Is it still true?
  • f. Multiply both sides of the (original) equation by −3−3 size 12{ - 3} {}. ___________ Is it still true?
  • g. Divide both sides of the (original) equation by −2−2 size 12{ - 2} {}. ___________ Is it still true?
  • h. In general: what operations, when performed on an inequality, reverse the inequality?

Exercise 2

2x + 3 < 7 2x + 3 < 7 size 12{2x+3<7} {}

  • a. Solve for x x.
  • b. Draw a number line below, and show where the solution set to this problem is.
  • c. Pick an x−value x value which, according to your drawing, is inside the solution set. Plug it into the original inequality 2x+3<72x+3<7 size 12{2x+3<7} {}. Does the inequality hold true?
  • d. Pick an x-value which, according to your drawing, is outside the solution set. Plug it into the original inequality 2x+3<72x+3<7 size 12{2x+3<7} {}. Does the inequality hold true?

Exercise 3

10 − x ≥ 4 10 − x ≥ 4 size 12{"10" - x >= 4} {}

  • a. Solve for xx size 12{x} {}. Your first step should be adding xx size 12{x} {} to both sides, so in your final equation, xx size 12{x} {} is on the right side.
  • b. Solve for xx size 12{x} {} again from the original equation. This time, leave xx size 12{x} {} on the left side.
  • c. Did your two answers come out the same?
  • d. Draw a number line, and show where the solution set to this problem is.
  • e. Pick an xx size 12{x} {}-value which, according to your drawing, is inside the solution set. Plug it into the original inequality 10−x≥410−x≥4 size 12{"10" - x >= 4} {}. Does the inequality hold true?
  • f. Pick an xx size 12{x} {}-value which, according to your drawing, is outside the solution set. Plug it into the original inequality 10−x≥410−x≥4 size 12{"10" - x >= 4} {}. Does the inequality hold true?

Exercise 4

x = ± 4 x = ± 4 size 12{x= +- 4} {}

  • a. Rewrite this statement as two different statements, joined by “and” or “or.”
  • b. Draw a number line, and show where the solution set to this problem is.

Exercise 5

− 3 < x ≤ 6 − 3 < x ≤ 6 size 12{ - 3<x <= 6} {}

  • a. Rewrite this statement as two different statements, joined by “and” or “or.”
  • b. Draw a number line, and show where the solution set to this problem is.

Exercise 6

x > 7 x > 7 size 12{x>7} {} or x < − 3 x < − 3 size 12{x< - 3} {}

Draw a number line, and show where the solution set to this problem is.

Exercise 7

x > 7 x > 7 size 12{x>7} {} and x < − 3 x < − 3 size 12{x< - 3} {}

Draw a number line, and show where the solution set to this problem is.

Exercise 8

x < 7 x < 7 size 12{x<7} {} or x > − 3 x > − 3 size 12{x> - 3} {}

Draw a number line, and show where the solution set to this problem is.

Exercise 9

x > ± 4 x > ± 4 size 12{x> +- 4} {}

  • a. Rewrite this statement as two different statements, joined by “and” or “or.”
  • b. Draw a number line below, and show where the solution set to this problem is.

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