Graphing an inequality on the TI-86

Note that the TI-86 and the TI-85 have slightly different keys. This page uses the keys associated with the TI-86. The differences are in the "2^nd" functions on some of the keys used here. The TI-85 keys will have the same key-face symbol unless otherwise noted.

The graphical solution to Example 14 Chapter 1 Section 2 of the text is not quite as simple to construct as one might hope. That example uses the problem

5x + 2 < x – 6 The solution shown in the text involves graphing each side of the inequality as a function, and then comparing the two function graphs. This page starts by constructing that solution. Then this page goes on to demonstrate graphing the entire inequality exactly as it is given. Below is a sequence of steps needed to generate that graph on a TI-86.

Figure 1	Figure 1 shows the result of pressing the key on a TI-86 calculator. The actual output depends on the previous state of the calculator. In this example, it would appear that no functions have been defined but that the window (i.e., range) settings are not at all standard.
Figure 2	Press the key to see the "y(x)=" list. On this calculator that list is empty. The calculator is ready to receive the first function definition.
Figure 3	To produce Figure 3 we have pressed the and keys to create the first function. Then we press the key to move to the second function. That function is created by the and keys.
Figure 4	Having entered the functions in Figure 3, we press to select the GRAPH command from the top menu. The result is shown in Figure 4. We can see the graphs of the two straight lines in Figure 4, but this graph does not appear as does the graph in the text. We will look at the WINDOW settings to see the current values and to change those values if need be.
Figure 5	To open the WINDOW menu we press the key to select the WIND command (RANGE on a TI-85). The result is shown in Figure 5. Clearly these are not the values that we want to be using. Rather than type all new values, let us use the ZOOM menu to change the WINDOW settings. To do this, press the key to select the ZOOM option. This will open a submenu, as is shown in Figure 6.
Figure 6	Figure 6 shows the start of the ZOOM submenu. We will opt for the ZSTD option by pressing . This will set the xMin value to ^–10, xMax to 10, xScl to 1, yMin to ^–10, yMax to 10, and yScl to 1. It also moves us back to graph mode and a new graph, see Figure 7, will be drawn.
Figure 7	In Figure 7 we have the new graph that uses the new WINDOW settings. It still does not seem to correspond to the figure in the book. First, the x-axis in Figure 7 needs to be raised. We can do this by making the yMin value more negative. Second, the "tick" marks on the y-axis are too close together. We can alter this by increasing the value assigned to yScl.
Figure 8	We move to Figure 8 by presing the key to open the WINDOW settings. A review of Figure 8 shows that we have indeed established the ZSTD settings described above.
Figure 9	Figure 9 shows the WINDOW settings screen after we have used the cursor keys to move down to the yMin line, and then we entered a new value for yMin, namely, ^–20. Then we moved to the yScl line and changed that value to be 2.
Figure 10	Now, to get to Figure 10 we press the key to do another graph. This graph looks similar to the graph in the textbook. However, the graph in the text tells us the x and y coordinate of the point of intersection of the two lines. We need to find the command to get the calculator to find that point. A review of the menu at the bottom of Figure 10 does not give any hint of our desired command. However, the small arrow at the right end of the menu indicates that there are more items. We press the key to see additional items in the menu.
Figure 11	Figure 11 shows the next five items in the menu. Now we have a interesting choice, namely, the MATH submenu. Press to select that choice.
Figure 12	The MATH submenu is displayed in Figure 12. Again, it does not seem to have the command that we desire. We press the key to see additional items in the MATH submenu. These can be seen in Figure 13.
Figure 13	The middle option in the MATH submenu is ISECT, a plausible abbreviation for INTERSECTION. We press the key to select the ISECT option.
Figure 14	As a result of selecting the ISECT option, Figure 14 shows us that the calculator is proposing that our line for y(x)=5x+2 be our first curve. We can examine the graph in Figure 14 and we will see that at the point, x=0, y=2, the calculator has displayed a special symbol. That is how the calculator identifies the particular line that it is proposing to use. We can press the key to accept that proposal. This will move the graph to Figure 15.
Figure 15	Now we need to select the second curve. The calculator is proposing the other line, y(x)=x-6, for the second curve. The calculator is making this proposal by displaying its flashing sign on a point, x=0, y=-6, on that line. Again, we will accept the proposal by pressing the key.
Figure 16	In Figure 16 the calculator has been given the two curves, and now it wants a guess, a starting point. In fact, the calcualtor offers the same point (0,-6) as that starting point. We can accept that point as our guess by pressing the key.
Figure 17	As a result of all of our efforts, the calculator display shifts to Figure 17. In that Figure, the calculator has identified the point of intersection of the two lines, namely at the point (^–2,^–8). This graph is remarkably similar to the graph that is given in the textbook. The only difference is that the textbook version contains the equations of the two lines. A closer examination of the graph in the textbook shows that these equations were pasted into the book; after all, the equations in the book appear in a completely different font. We too can doctor a picture. We have done that in Figure 18.
Figure 18	You can not produce Figure 18 directly from the calculator. The equations have been added to this image. In addition, we have added a red line from the point of intersection straight up to the x-axis. In additon, we have replaced the x-axis from directly above the point of intersection all the way to the left by a light blue line. This represents the x-values where the line y(x)=5x+2 is below the line y(x)=x–6. In effect, this is the set of points where 5x+2 < x–6 which was the original problem.

It might have been nice to be able to graph the original problem

5x + 2 < x – 6 more directly. We can do this on the TI-86 (and on the TI-85) by observing that these calculators represent the value of "TRUE" with the number 1 and the value "FALSE" with the number 0.

Figure 19 Figure 19 returns to the "y=" screen. The key returned the main graph menu to Figure 18. Then the key will open the "y=" screen. For this example we have deleted any existing function definitions in that screen, using the key and the cursor control keys. Then we returned to the y1= line where we start our entry of 5x + 2 < x – 6 by pressing . That leaves us with the challenge of producing the "less than" sign, which does not appear on the keyboard.

Figure 20 To produce the "less than" sign we press the keys to open the TEST menu shown in Figure 20. This menu gives us the option of choosing the "less than" character.

Figure 21 Figure 21 completes the function by pressing to select the "less than" character from the menu, and then to finish the line.
At this point we note that the option to GRAPH has been lost from the menu scheme.

Figure 22 We press to close the bottom menu of Figure 21 and therefore to produce Figure 22.

Figure 23 Rather than just make the graph of our function, we will use the ZOOM menu options to change the WINDOW settings. Therefore, we use the keys to change the menu structure to that shown in Figure 23. Then, we press to select the STANDARD settings and move immediately to Figure 24.

Figure 24 Figure 24 shows the graph of the the inequality, where every x-value that makes the inequality true is associated with the y-value 1, and every x-value that makes the inequlity false is associated with the y-value 0. Unfortunately, graphing the y-value 0 merely puts the graph on top of the x-axis. This makes it hard to see.
We can modify the orginal function to improve the visibility of the function.

Figure 25 We return to the y= screen via the key.
We know that the expression
5x+2 < x–6 has the value 0 or 1 (representing FALSE or TRUE). If we multiply that expression by 3 and then subtract 1 we will have an expression that evaluates as ^–1 or 2. Our new expression is (5x+2 < x–6)*3–1 If we place the cursor at the start of the expression, on top of the 5, then pressing will shift the calculator into insert mode. Then, press the to generate the initial left parenthesis.
Then move to the right end of the expression by pressing the key, and finish the expression with .

Figure 26 Pressing moves the calculator from Figure 25 to Figure 26. Here we can see just where the expression is true, i.e., where it has the value 2, and where the expression is false, i.e., has the value ^–1.

Figure 27 Knowing that the graph ranges from 2 to ^–1, we can return to the WINDOW screen, via the key, where we can change the values for yMin and yMax. Figure 27 shows the screen with the settings generated by our ZOOM Standard, back in Figure 23.

Figure 28 In Figure 28 we have changed the settings. After making the appropriate changes, we can press to move back to the graph, as shown in Figure 29.

Figure 29 The graph has changed as a result of our changes in the yMin and yMax values. It is now even easier to see the change from true (the value 2) to false (the value ^–1). At the same time the almost vertical line connecting the true values to the falsw values has become even more prominent. That nearly vertical line is not part of the graph. At no time does (5x+2 < x–6)*3–1 produce any value other than 2 or ^–1. The nearly vertical line is a result of the calculator setting that attempts to vertically connect adjacent portions of the graph. We can turn off that setting.

Figure 30 To turn off the setting we need to move to the FORMT command in the menu. FORMT is not shown in the menu of Figure 29. However, if we press the key, the menu changes to that shown in Figure 30. Now we can use the FORMT command by pressing the key.

Figure 31 The FORMT command opens the window shown in Figure 31. We are interested in the third line where we have two settings, DrawLine and DrawDot. The setting shown in Figure 31 indicates that the calculator is in the DrawLine mode, since that is the option that is highlighted. We use the cursor keys to move down to the third line and then over to the DrawDot option. Once there, we press the key to select that option. The result is shown in Figure 32.

Figure 32 Now that we have changed the option to DrawDot, we can press to graph the function again, but this time with the new option set.

Figure 33 Figure 33 shows the new version of the graph. Note that there is no longer a connecting line between the true values and the false values.

Figure 19	Figure 19 returns to the "y=" screen. The key returned the main graph menu to Figure 18. Then the key will open the "y=" screen. For this example we have deleted any existing function definitions in that screen, using the key and the cursor control keys. Then we returned to the y1= line where we start our entry of 5x + 2 < x – 6 by pressing . That leaves us with the challenge of producing the "less than" sign, which does not appear on the keyboard.
Figure 20	To produce the "less than" sign we press the keys to open the TEST menu shown in Figure 20. This menu gives us the option of choosing the "less than" character.
Figure 21	Figure 21 completes the function by pressing to select the "less than" character from the menu, and then to finish the line. At this point we note that the option to GRAPH has been lost from the menu scheme.
Figure 22	We press to close the bottom menu of Figure 21 and therefore to produce Figure 22.
Figure 23	Rather than just make the graph of our function, we will use the ZOOM menu options to change the WINDOW settings. Therefore, we use the keys to change the menu structure to that shown in Figure 23. Then, we press to select the STANDARD settings and move immediately to Figure 24.
Figure 24	Figure 24 shows the graph of the the inequality, where every x-value that makes the inequality true is associated with the y-value 1, and every x-value that makes the inequlity false is associated with the y-value 0. Unfortunately, graphing the y-value 0 merely puts the graph on top of the x-axis. This makes it hard to see. We can modify the orginal function to improve the visibility of the function.
Figure 25	We return to the y= screen via the key. We know that the expression 5x+2 < x–6 has the value 0 or 1 (representing FALSE or TRUE). If we multiply that expression by 3 and then subtract 1 we will have an expression that evaluates as ^–1 or 2. Our new expression is *(5x+2 < x–6)3–1** If we place the cursor at the start of the expression, on top of the 5, then pressing will shift the calculator into insert mode. Then, press the to generate the initial left parenthesis. Then move to the right end of the expression by pressing the key, and finish the expression with .
Figure 26	Pressing moves the calculator from Figure 25 to Figure 26. Here we can see just where the expression is true, i.e., where it has the value 2, and where the expression is false, i.e., has the value ^–1.
Figure 27	Knowing that the graph ranges from 2 to ^–1, we can return to the WINDOW screen, via the key, where we can change the values for yMin and yMax. Figure 27 shows the screen with the settings generated by our ZOOM Standard, back in Figure 23.
Figure 28	In Figure 28 we have changed the settings. After making the appropriate changes, we can press to move back to the graph, as shown in Figure 29.
Figure 29	The graph has changed as a result of our changes in the yMin and yMax values. It is now even easier to see the change from true (the value 2) to false (the value ^–1). At the same time the almost vertical line connecting the true values to the falsw values has become even more prominent. That nearly vertical line is not part of the graph. At no time does *(5x+2 < x–6)3–1** produce any value other than 2 or ^–1. The nearly vertical line is a result of the calculator setting that attempts to vertically connect adjacent portions of the graph. We can turn off that setting.
Figure 30	To turn off the setting we need to move to the FORMT command in the menu. FORMT is not shown in the menu of Figure 29. However, if we press the key, the menu changes to that shown in Figure 30. Now we can use the FORMT command by pressing the key.
Figure 31	The FORMT command opens the window shown in Figure 31. We are interested in the third line where we have two settings, DrawLine and DrawDot. The setting shown in Figure 31 indicates that the calculator is in the DrawLine mode, since that is the option that is highlighted. We use the cursor keys to move down to the third line and then over to the DrawDot option. Once there, we press the key to select that option. The result is shown in Figure 32.
Figure 32	Now that we have changed the option to DrawDot, we can press to graph the function again, but this time with the new option set.
Figure 33	Figure 33 shows the new version of the graph. Note that there is no longer a connecting line between the true values and the false values.