Chapter 1, Section 2, Example 16 on TI-89

This page looks at Example 16 from Chapter 1 Section 2. The problem in that example is

^–7 2x + 5 8 This is a "combined inequality". As noted in the text and on the combined inequality web page, this problem is merely a shorthand version of ^–7 2x + 5 and 2x + 5 8 For reasons noted in the the combined inequality web page, we can not ask the calculator to use the "combined inequality". However, we can use the calculator to obtain a graph of the expanded version.

Figure 1 We start entering the inequality on the "y=" screen. The keys open the "y=" screen shown in Figure 1.
The TI-89 evaluates inequalities as TRUE or FALSE. Furthermore, these are "logical" or "boolean" values, and on the TI-89 there is no numeric representation for such values. The function "tester()" has been written to evaluate such "boolean" expressions and to produce a 1 if the expression is TRUE and a 0 if the expression is FALSE. Once it has been loaded onto the calculator, we can find the "tester()" function by pressing the keys to open the window shown in Figure 2.

Figure 2 Figure 2 shows the start of the "VAR-LINK" window on this calculator. The contents of the window will change on each calculator as new variables, programs, and functions are defined on the calculator. We could use the cursor keys to move down the list of variables, programs, and functions looking for the tester() function. However, pressing the key will move the display to the next entry that starts with the letter "t".

Figure 3 We seem to be in luck on this calculator. The highlight has moved to the desired "tester" function.

Figure 4 We select that highlighted item from Figure 3 and paste it into the function entry and edit line by pressing the key. The result is shown in Figure 4.

Figure 5 First we produce the openning quotation mark via the keys. Then we start to construct the inequality inside quotation marks. produces "(^–7". The next character that we need to enter is the "less than or equal to" character. We can do this via the keys. And we can finish the first portion of the inequality with the keys.

Figure 5a At this point we need to generate the " and " for our statement. There are many ways to do this. One way would be to enter the spaces and the letters using the keyboard in alphabetic mode. Another way would be to open the CATALOG and select the " and " from there. A third method is to use the MATH menu. We will demonstrate that third method.
Press to open the MATH menu shown in Figure 5a. From this menu we want to select the "Test" option. Therefore, press the key to select that option and move to Figure 5b.

Figure 5b Here, in the Test sub-menu, we can see that "and" is the eighth choice. Thereofre, we press the key to make that selection.

Figure 5c The calculator has taken our choice and has pasted the " and " at the end of our function.

Figure 6 Now we can complete the function via .

Figure 7 Pressing the key accepts the function, placing it in the top portion of the window.
In preparation for actually graphing the function we move to the ZOOM window by pressing . The result is shown in Figure 8.

Figure 8 In Figure 8 we have the various Zoom options. We will select ZoomStd by pressing . This will move us to the graph window shown in Figure 9.

Figure 9 Figure 9 demonstates the solution, the half-closed, half-opened interval [^–6,3/2). (Note that the appearance of the graph in Figure 9 is dependent on the WINDOW being in the ZoomStd settings.) Although we will not see the fine distinctions at the ends of the interval, the graph is helpful in confirming the location of the solution.

As noted on other web pages, the graph shown above, although correct and although it does a good job of showing where the inequality is TRUE, does not do a good job of showing where the inequality is FALSE. This is so because the value FALSE is associated with 0 by the tester() function on the TI-89 and as the calculator plots those 0 values the graph is on top of the x-axis. Therefore, we do not see any change to the graph as the 0 values are plotted. The scheme that we have used before to emphasize the TRUE and FALSE values is to multiply the entire function by 3 and then subtracting 1. As a result, TRUE is plotted as 2 while FALSE is plotted as ^–1. The following images show the conversion of the formula and the resulting graph.

Figure 10 First we will prepare the way for our final graph by adjusting the WINDOW settings. We can move to Figure 10 by pressing the keys, and then using the cursor keys to move down and to change yMin to ^–3, ymax to 3, and xres to 1.

Figure 11 The keys move the display back to the graph. We can see the consequences of our changes in Figure 11.
We need to move back to the "y=" screen to make the changes to the function noted above. Therefore, we press to move to Figure 12.

Figure 12 In Figure 12 we are back in the "y=" screen. The highlight is on the "y2=" line. We want to edit the function "y1=". Therefore, we use the key to move the highlight.

Figure 13 In Figure 13 we have the highlight on the function that we want to edit. However, we need to press the key to move the highlight down to the function entry and edit line.

Figure 14 We want to multiply this function by 3 and then subtract 1. Do do this we want to move to the right end of the function. Because the enitre function is highlighted in Figure 14, we can press the key once to move to the right end of the function.

Figure 15 In Figure 15 we are finally in place to make the desired change.

Figure 16 We generate the change by pressing the keys.
To accept the change we could press the ENTER key. However, our desire is to accept the change and move directly to the graph. We can do this by pressing the keys.

Figure 17 Figure 17 displays the changed function. On this graph we can see clearly the area where the inequality is FALSE (resulting in 0*3–1 or ^–1) and where it is TRUE (resulting in 1*3–1 or 2).

Figure 1	We start entering the inequality on the "y=" screen. The keys open the "y=" screen shown in Figure 1. The TI-89 evaluates inequalities as TRUE or FALSE. Furthermore, these are "logical" or "boolean" values, and on the TI-89 there is no numeric representation for such values. The function "tester()" has been written to evaluate such "boolean" expressions and to produce a 1 if the expression is TRUE and a 0 if the expression is FALSE. Once it has been loaded onto the calculator, we can find the "tester()" function by pressing the keys to open the window shown in Figure 2.
Figure 2	Figure 2 shows the start of the "VAR-LINK" window on this calculator. The contents of the window will change on each calculator as new variables, programs, and functions are defined on the calculator. We could use the cursor keys to move down the list of variables, programs, and functions looking for the tester() function. However, pressing the key will move the display to the next entry that starts with the letter "t".
Figure 3	We seem to be in luck on this calculator. The highlight has moved to the desired "tester" function.
Figure 4	We select that highlighted item from Figure 3 and paste it into the function entry and edit line by pressing the key. The result is shown in Figure 4.
Figure 5	First we produce the openning quotation mark via the keys. Then we start to construct the inequality inside quotation marks. produces "(^–7". The next character that we need to enter is the "less than or equal to" character. We can do this via the keys. And we can finish the first portion of the inequality with the keys.
Figure 5a	At this point we need to generate the " and " for our statement. There are many ways to do this. One way would be to enter the spaces and the letters using the keyboard in alphabetic mode. Another way would be to open the CATALOG and select the " and " from there. A third method is to use the MATH menu. We will demonstrate that third method. Press to open the MATH menu shown in Figure 5a. From this menu we want to select the "Test" option. Therefore, press the key to select that option and move to Figure 5b.
Figure 5b	Here, in the Test sub-menu, we can see that "and" is the eighth choice. Thereofre, we press the key to make that selection.
Figure 5c	The calculator has taken our choice and has pasted the " and " at the end of our function.
Figure 6	Now we can complete the function via .
Figure 7	Pressing the key accepts the function, placing it in the top portion of the window. In preparation for actually graphing the function we move to the ZOOM window by pressing . The result is shown in Figure 8.
Figure 8	In Figure 8 we have the various Zoom options. We will select ZoomStd by pressing . This will move us to the graph window shown in Figure 9.
Figure 9	Figure 9 demonstates the solution, the half-closed, half-opened interval [^–6,3/2). (Note that the appearance of the graph in Figure 9 is dependent on the WINDOW being in the ZoomStd settings.) Although we will not see the fine distinctions at the ends of the interval, the graph is helpful in confirming the location of the solution.

Figure 10	First we will prepare the way for our final graph by adjusting the WINDOW settings. We can move to Figure 10 by pressing the keys, and then using the cursor keys to move down and to change yMin to ^–3, ymax to 3, and xres to 1.
Figure 11	The keys move the display back to the graph. We can see the consequences of our changes in Figure 11. We need to move back to the "y=" screen to make the changes to the function noted above. Therefore, we press to move to Figure 12.
Figure 12	In Figure 12 we are back in the "y=" screen. The highlight is on the "y2=" line. We want to edit the function "y1=". Therefore, we use the key to move the highlight.
Figure 13	In Figure 13 we have the highlight on the function that we want to edit. However, we need to press the key to move the highlight down to the function entry and edit line.
Figure 14	We want to multiply this function by 3 and then subtract 1. Do do this we want to move to the right end of the function. Because the enitre function is highlighted in Figure 14, we can press the key once to move to the right end of the function.
Figure 15	In Figure 15 we are finally in place to make the desired change.
Figure 16	We generate the change by pressing the keys. To accept the change we could press the ENTER key. However, our desire is to accept the change and move directly to the graph. We can do this by pressing the keys.
Figure 17	Figure 17 displays the changed function. On this graph we can see clearly the area where the inequality is FALSE (resulting in *03–1 or ^–1) and where it is TRUE (resulting in 13–1* or 2).