Title
Multiple Inequalities

Problem Statement
Find as many inequalities as possible of the form ax + b > c whose solutions are x < 12.

Problem setup

What are some possible inequalities with a solution set x < 12?  What happens if x is positive?  Negative?

Plans to Solve/Investigate the Problem

We will use the graphing calculator program to enter the two formulas x < 12 and ax + b > c and show where the overlapping area is the solution to the problem.

Investigation/Exploration of the Problem

In the expression ax + b > c, when we solve for x, we find that if x > (c-b)/a.  Therefore,  when a is negative, then x > (c-b)/a.  If a is positive, then (c-b)/a < x.  By graphing the two formulas, we can show the solution to be the overlapping shaded area.  We set the values of a = -1, b=1, and c=5.  In this case, we find the solution is the set of numbers between -4 and 12.

Purple

c=5

b=1                                                                                                                                          a=-1

Red

Extensions of the Problem

When reflecting on the pedagogy of this problem, we find that for teaching middle graders, we could extend the problem by showing students the combination of the two inequalities x > (c-b)/a and x < 12, and then graphing the result.  This would provide a clearer image and understanding of the inequality (c-b)/a < x < 12 as two different inequalities.  By using the same values for a, b, and c as above, the solution would be the same numbers -4 to 12 and would look like this:

c = 5

b = 1

a = -1

Further, this problem might be used to demonstrate why the inequality sign is reversed in an inequality divided or multiplied by a negative number when solving for x.

As an extension of the problem, we might demonstrate why it is necessary to reverse the inequality sign when multiplying or dividing by a negative number when solving for x.

Author & Contact
Jill Jackson and Shirley Crawford
jjackson@rockdale.k12.ga.us or scrawford@rockdale.k12.ga.us

Further, this problem might be used to demonstrate why the inequality sign is reversed in an inequality divided or multiplied by a negative number when solving for x.

Further, this problem might be used to demonstrate why the inequality sign is reversed in an inequality divided or multiplied by a negative number when solving for x.