Chapter 1, Section 2, Example 16 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.

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, and produces the image shown in Figure 1.
The next character that we need to enter is the "less than or equal to" character. To do this we will open the TEST menu via the keys. The new menu is displayed at the bottom of Figure 2.

Figure 2 selects the from the fourth position in the sub-menu. Then we can complete Figure 2 via the keys.
At this point we want to continue the inequality with "and". We could actually type the word, being careful to do so in lower case letters, but it might be easier to find "and" in the CATALOG.

Figure 3 In Figure 3 we have opened the CATALOG via the keys. Then, we can move to the items in the CATALOG by pressing the key. That will take us to Figure 4.

Figure 4 The CATALOG on this calculator opened at the top of its list. (Had the CATALOG opened at some other point we could have pressed the key to move directly to the "a's".) In Figure 4, the "abs" entry is selected by the arrow at the left of the screen.

Figure 5 We press the key tomove the selection arrow down to the "and" entry. Then, press to actually select that item.

Figure 6 Note that after our use of the CATALOG we have been returned to the "y=" screen and that "and" has been pasted into our expression. Now we are ready to complete our problem.

Figure 7 generates the repeat of the middle expression from the combined inequality.

Figure 8 To complete the line we press to open the TEST menu, and then press .

Figure 9 The key closes the bottom menu of Figure 8, changing the menu structure to that shown in Figure 9. Then, will select the GRAPH option from the top menu, and move us to Figure 10.

Figure 10 Figure 10 demonstates the solution, the half-closed, half-opened interval [^–6,3/2). (Note that the appearance of the graph in Figure 10 is dependent on the WINDOW being in the ZOOM Standard 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 on the TI-86 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 inequality 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 11 In Figure 11, the key has returned the screen to the "y=" window. There we have entered "insert" mode via the keys, and we have inserted a left parenthesis, , at the start of the expression.

Figure 12 moves the cursor to the right end of the expression. There completes Figure 12.

Figure 13 Knowing that our function will produce either 2 or ^–1 we can narrow the range of Y values by using the keys to select the WIND option from the top menu of Figure 12. This will open the WINDOW screen shown in Figure 13. here we can see that the yMin value is the default ^–10, and the yMax value is 10. We can change these values.

Figure 14 For Figure 14 we have used the cursor keys to move around in the screen and we have changed the yMin and yMax values. Pressing the key will move us to Figure 15.

Figure 15 Here the graph is drawn. Now it is easier to see both the TRUE and the FALSE regions.

Figure 1	We start entering the inequality on the "y=" screen. The keys open the "y=" screen, and produces the image shown in Figure 1. The next character that we need to enter is the "less than or equal to" character. To do this we will open the TEST menu via the keys. The new menu is displayed at the bottom of Figure 2.
Figure 2	selects the from the fourth position in the sub-menu. Then we can complete Figure 2 via the keys. At this point we want to continue the inequality with "and". We could actually type the word, being careful to do so in lower case letters, but it might be easier to find "and" in the CATALOG.
Figure 3	In Figure 3 we have opened the CATALOG via the keys. Then, we can move to the items in the CATALOG by pressing the key. That will take us to Figure 4.
Figure 4	The CATALOG on this calculator opened at the top of its list. (Had the CATALOG opened at some other point we could have pressed the key to move directly to the "a's".) In Figure 4, the "abs" entry is selected by the arrow at the left of the screen.
Figure 5	We press the key tomove the selection arrow down to the "and" entry. Then, press to actually select that item.
Figure 6	Note that after our use of the CATALOG we have been returned to the "y=" screen and that "and" has been pasted into our expression. Now we are ready to complete our problem.
Figure 7	generates the repeat of the middle expression from the combined inequality.
Figure 8	To complete the line we press to open the TEST menu, and then press .
Figure 9	The key closes the bottom menu of Figure 8, changing the menu structure to that shown in Figure 9. Then, will select the GRAPH option from the top menu, and move us to Figure 10.
Figure 10	Figure 10 demonstates the solution, the half-closed, half-opened interval [^–6,3/2). (Note that the appearance of the graph in Figure 10 is dependent on the WINDOW being in the ZOOM Standard 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 11	In Figure 11, the key has returned the screen to the "y=" window. There we have entered "insert" mode via the keys, and we have inserted a left parenthesis, , at the start of the expression.
Figure 12	moves the cursor to the right end of the expression. There completes Figure 12.
Figure 13	Knowing that our function will produce either 2 or ^–1 we can narrow the range of Y values by using the keys to select the WIND option from the top menu of Figure 12. This will open the WINDOW screen shown in Figure 13. here we can see that the yMin value is the default ^–10, and the yMax value is 10. We can change these values.
Figure 14	For Figure 14 we have used the cursor keys to move around in the screen and we have changed the yMin and yMax values. Pressing the key will move us to Figure 15.
Figure 15	Here the graph is drawn. Now it is easier to see both the TRUE and the FALSE regions.