*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

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.