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Graphing Inequalities and Absolute Values

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

Summary: This module provides sample problems which develop concepts related to graphing inequalities and absolute values.

Exercise 1

9x + 3y ≤ 6 9x + 3y ≤ 6 size 12{9x+3y <= 6} {}

  • a. Put into a sort of y=mx+by=mx+b size 12{y= ital "mx"+b} {} format, except that it will be an inequality.
  • b. Now, ignore the fact that it is an inequality—pretend it is a line, and graph that line.
  • c. Now, to graph the inequality, shade in the area either above the line, or below the line, as appropriate.

    Hint:

    Does yy size 12{y} {} have to be less than the values on the line, or greater than them?
  • d. Test your answer. Choose a point (any point) in the region you shaded, and test it in the inequality. Does the inequality work? (Show your work.)
  • e. Choose a point (any point) in the region you did not shade, and test it in the inequality. Does the inequality work? (Show your work.)

Exercise 2

4x ≤ 2y + 5 4x ≤ 2y + 5 size 12{4x <= 2y+5} {}

  • a. Graph the inequality, using the same steps as above.
  • b. Test your answer by choosing one point in the shaded region, and one point that is not in the shaded region. Do they give you the answers they should? (Show your work.)

Exercise 3

y = ∣ x ∣ y = ∣ x ∣ size 12{y= lline x rline } {}

  • a. Create a table of points. Your table should include at least two positive xx size 12{x} {}-values, two negative xx size 12{x} {}-values, and x=0x=0 size 12{x=0} {}.
  • b. Graph those points, and then draw the function.

Exercise 4

y=∣x∣+3y=∣x∣+3 size 12{y= lline x rline +3} {}. Graph this without a table of points, by remembering what “adding 3” does to any graph. (In other words, what will these yy size 12{y} {}-values be like compared to your yy size 12{y} {}-values in #3?)

Exercise 5

y=−∣x∣y=−∣x∣ size 12{y= - lline x rline } {}. Graph this without a table of points, by remembering what “multiplying by –1” does to any graph. (In other words, what will these yy size 12{y} {}-values be like compared to your yy size 12{y} {}-values in #3?)

Exercise 6

Now, let’s put it all together!!!

  • a. Graph y=−∣x∣+2y=−∣x∣+2 size 12{y= - lline x rline +2} {}.
  • b. Graph y≥∣x∣+2y≥∣x∣+2 size 12{y >= = lline x rline +2} {}. Your answer will either be a shaded region on a 2-dimensional graph, or on a number line.
  • c. Test your answer by choosing one point in the shaded region, and one point that is not in the shaded region. Do they give you the answers they should? (Show your work.)
  • d. Graph –|x|+2 < 0. Your answer will either be a shaded region on a 2-dimensional graph, or on a number line.
  • e. Test your answer by choosing one point in the shaded region, and one point that is not in the shaded region. Do they give you the answers they should? (Show your work.)

Exercise 7

Extra for experts:

y≥3∣x+4∣y≥3∣x+4∣ size 12{y >= 3 lline x+4 rline } {}

  • a. Graph it. Think hard about what that +4 and that 3 will do. Generate a few points if it will help you!
  • b. Test your answer by choosing one point in the shaded region, and one point that is not in the shaded region. Do they give you the answers they should? (Show your work.)

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