You can enter an inequality on the TI89 just as you would enter an equation. The "<" symbol is entered by pressing <2nd> "0", and the ">" symbol is entered by pressing <2nd>".". You may have already noticed these symbols on the calculator. It isn't shown on the calculator face, so you may not known that you can enter "" by pressing <Diamond>"0", and "" by pressing <Diamond>".". You can also access these operators using the Math Test menu (<2nd><5><8>).
Here are some screen shots of a solution to problem A1.26
(on p. A8) in the Larsen text, which says in part: "Solve
the inequality ." 

Next, type the inequality, ",", then the variable you wish to solve for, and ")" <ENTER>. The solution is in interval notation. 

This screen shot shows you how to access the inequality operators from the Math Test menu (<2nd><5><8>) instead of the keyboard, if you prefer. 

Here is a "manual" solution to the same problem. This method can be useful in locating an error in paperandpencil work. After typing the problem, the solution proceeds by subtracting 2x from both sides (type "", "2", "x", <ENTER>), then subtract 1 from both sides (type "", "1", <ENTER>  the "ans(1)" is supplied automatically by the calculator. 

Look at problem A1.28 on p. A8 of the Larsen text. It says, in part "solve the inequality "  let's pretend that it isn't trivial. The TI89 won't manipulate a "double inequality" like this directly, you need to separate it into two separate inequalities as shown on the right. Of course, you still have to interpret the answer. 
