While it is necessary that you be able to solve polynomial inequalities "by hand", it would be very handy to be able to use the TI89 to check your work or generate practice problems. Unfortunately, the TI89 manual states that the solve command can only be used to solve linear inequalities. However, there is a way to "trick" the solve command into solving polynomial and rational inequalities. (The userdefined functions below are based on a method described by Jinghuang Tian of Rio Salado Community College, Tempe, AZ in the May, 2002 Mathematics Teacher (Vol. 95, No. 5, p. 3845).)
Problem 44 on page A8 of the Larsen text is:
"Solve the inequality 2x^{2} + 1 < 9x  3 and graph the solutions on the real line."
First, you can use the "solve(" command to solve the related equation. To do this, after selecting NewProb from the "F6  Clean Up" menu, select "1: solve(" from the "F2  Algebra" menu. 

Next, enter the related equation "2x^{2} + 1 = 9x  3" and solve it for x. 

The solutions divide the number line into three regions, x < 1/2, 1/2 < x < 4, and x > 4. Pick a convenient value in each interval and see if it satisfies the original inequality. If it does, all values of x in the region will also satisfy the inequality. You can use the "" key to substitute a value, as shown at right. (I have chosen x = 0 from the lefthand region, x = 1 from the middle region, and x = 5 from the righthand region. The calculation "2x^{2} + 1 < 9x  3  x = 0" has scrolled off the screen. Since the inequality is true for x = 1, the solution to the inequality is 1/2 < x < 4. 

The easy way to solve problems such as this on the TI89 is to create a userdefined function to do the job. Once this function is created, it will remain in the calculator's memory. The function is:
Define solvepi(f,x,s) = solve(sign(factor(f,x))=s,x)
Here is how to create this function in your TI89:
After starting a new problem, press <F4> and select "1: Define". 

Every function has a name. I have chosen the name "solvepi", which stands for "solve polynomial inequalities". (It is a very good idea to make the names of functions mnemonic.) Type the name by pressing <2nd> <alpha> "solvepi" <alpha> 

Next, type the function parameters, which are values that you supply when you call the function. This function has three parameters: "f", the expression to evaluate, "x", the variable used in the expression, and "s", which takes the value 1 (for ">") or 1 (for "<"). To enter the parameters, type "(" <alpha> "f,x,", <alpha> "s)" 

Finally, the function's body needs to be entered. First, type "=", then press <F2><1> to select "solve(". 

To get the "sign(" function, you can select the math menu by pressing <2nd> <MATH>, then press <1> to select the Number submenu, then press <8> to select "sign(", or: 

You can press <CATALOG>, then "S", and use the arrow keys to scroll down to "sign(", or you can type the function by pressing <2nd><alpha> "sign" <alpha> "(". 

You can get the "factor(" function by pressing <F2> <2>, or by typing it. 

To finish the body of the function, type <alpha> "f,x))=" <alpha> "s,x)" and press <ENTER> 
To solve the inequality "2x^{2} + 1 < 9x  3", you can either select the function name from the VARLINK menu (press <2nd><VARLINK>, then "s"), or type it by pressing <2nd><alpha> "solvepi" <alpha> "(". 

The function takes three parameters:

How does the "solvepi" function work?
In our particular course, we don't generally solve rational inequalities, but the method described by Tian leads to the following userdefined function:
Define solveri(n, d, x, s) = solve(sign(factor(n,x)/factor(d,x))=s,x)
works to solve inequalities of the form n/d > 0 (s = 1) and n/d < 0 (s = 1). You are welcome to experiment with it if you like.