Objective:
To provide a visual tool to aid in understanding the meaning of solutions
to inequalities and solutions to systems of inequalities.
Level: Algebra
or Precalculus.
Prerequisites:
A
basic introduction to linear inequalities and some experience in solving
them by "hand". This should include solutions to equations in two unknowns
(in particular, that such a solution is an ordered pair) and terminology
like half-plane. If inequalities involving functions other than linear
functions (like quadratics) are to be explored, their properties and graphs
should have been discussed.
In a technology equipped classroom, the
interactive instructional tool discussed in this demo can be useful in
introducing the concepts involved in the solution of inequalities prior
to students solving them by hand. The visual components of this tool parallel
the traditional steps taught for hand calculations.
Platform: Matlab
or an accompanying JAVA
applet which performs simulations discussed in this demo:
javaineq.
Instructor's
Notes:
The basic scheme for solving linear inequalities
is obtained from the following observation. A line Ax + By = C divides
a plane into
three (disjoint) sets of points:
the set of points satisfying Ax + By < C,
Ax + By = C, and Ax + By > C. Hence
we are able to identify a three-step procedure to determine the solution
set of a single linear inequality. These steps can be performed by hand
or using appropriate computer or calculator tools.
1. Graph.
Graph the line corresponding to the linear inequality. This is
often done by first determining a line associated with the inequality,
putting it into the slope intercept form y = mx + b, generating
at
least two points on the this line, and connecting the points. A dashed
line is used to indicate a strict inequality and a solid line to indicate
that the line itself is included.
2. Test.
Next we test a point not on the line. If the graph does not pass
through the origin (0,0), then it is easily determined by inspection
whether (0,0) satisfies the inequality or not.
3. Shade.
Finally we ask that the student shade the half-plane that
consists of the points for which the inequality is true. If the inequality
was true at the test point, then the half-plane containing that point is
the part to shade.
For example the set of all points that
satisfy the inequality y > 2x - 4 is the shaded half-plane in Figure
1.
Figure 1.
To provide a visual tool for solving a
single inequality and to lay the foundations for solving a system of inequalities
the Matlab routine inequalities was developed. This routine follows
the three-step procedure described above and provides an opportunity to
extend the ideas beyond linear inequalities and to systems of inequalities.
An annotated screen is shown in Figure 2.
Figure 2.
The student uses this tool just as though
they were proceeding by hand. All the conceptual steps are reflected in
choices made by the user of the routine. Hence this is not a sequence of
button selections that automatically solves the inequality.
A major benefit of using a visual tool
as displayed in Figure 2 is for understanding solutions of systems of linear
inequalities. The process now requires the solution of the individual inequalities
followed by filling in the region where the individual solutions overlap.
The student can test each of the inequalities and locate the two solution
regions. Once the overlapping portion is determined the "Fill a Region"
button can be used to outline and then fill in the solution region of the
system.
The sequence of steps for solving a system
of linear inequalities is shown in the following animation which we label
as Figure 3. There is a time delay between frames so that you can read
the annotations which have been inserted.
Figure 3.
For a slower presentation which could be
part of an introduction to this visual tool, we make the following approach
available. The preceding animation consists of 13 frames. For instructional
purposes we have isolated these frames so they may be viewed individually.
Click on the description to see the corresponding instructional step for
solving a system of linear inequalities.
This visual tool can also be used to
solve systems of inequalities of the following type.
Find the region that satisfies
both y < (-1/4)x2 and y > -3x - 2.
Using the same strategy as for a system of
linear inequalities we obtain
Figure 4.
Figure 4.
A worksheet for single inequalities and
another for systems of inequalities is available by clicking on the following
boxes.
The MATLAB m-file described in this demo can
be downloaded by clicking on its name inequalities. We have also developed a
JAVA applet for this demo which has the
similar functionality :
javaineq.
(The displays of the applet may vary slightly because of the browser you use
and the screen resolution of your monitor.) This routine is highly interactive and has been used successfully in the
high school classroom for several years.
Other resources:
QuickMath (Automatic Math Solutions)
at http://www.quickmath.com/ has
a section that allows you to solve inequalities and systems of inequalities
in a single variable. There is also a plot feature. This site produces
answers, with no interactive steps for learning solution techniques.
From Addison Wesley Longman the following
is available. "MathNotes.com is
a valuable on-line resource for both instructors and student users of the
Lial/Hornsby/Miller Paperback Series. Teachers can find materials to enhance
their courses, while students can strengthen their understanding through
interactive tutorials and study aids, or, they can more deeply explore
concepts through real world applications and Web links for further research."
Click on a link to Intermediate Algebra and you will find links to material
on inequalities.
At http://www.exploremath.com/activities/activity_list.cfm?categoryID=8
use keyword inequalities to see a group of gizmos for investigating different types of
inequalities including single linear inequalities and systems of linear
inequalities. The interface is very nicely done and uses sliders to change
the value of slopes any y-intercepts. Unfortunately there is no reinforcement
of the steps for solving inequalities. Shockwave™ plugin required to view
multimedia activities. Exploremath is a comprehensive site for mathematics
instructional tools.
The preceding list is not exhaustive and
only gives an indication of options that are available for teaching inequalities
as described in this demo. Using a search engine you can locate other resources.
A site that is continuing to update a list of resources for mathematics
is http://archives.math.utk.edu/.
Credits:
This demo was submitted by
Tommy
Vizza
Mathematics Faculty
Woodlynde School
and is included in Demos
with Positive Impact with his permission. The Java applet was developed by Philip Nicastro
a student at Temple University.
