Inequalities

 Solution to an inequality.
 
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

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.
  
                             1. Introduce the problem.
2. Select a symbol.    7. Select second symbol.
3. Enter first expression. 8. Enter second expression.
4. Graph first line.  9. Graph second line.
5. Test a region.  10. Test a region.
6. Shade a region.   11. Select to Fill a region button.
    Go to second column.  12. Outline overlapping TRUE region.
13. Shade the solution set.


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. 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 there are a group of on-line tools 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 rather comprehensive site for mathematics instructional tools.


A number of high school algebra books indicate software support for teaching inequalities. Several of these mention Graph Explorer. A java applet of Graph Explorer is available at www.langara.bc.ca/mathstats/resource/GraphExplorer/

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.


 
                                                                                   DRH 3/19/01