# on the TI-89

## Introduction:

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 TI-89 to check your work or generate practice problems. Unfortunately, the TI-89 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 user-defined 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. 384-5).)

## Polynomial Inequalities

Problem 44 on page A8 of the Larsen text is:

"Solve the inequality 2x2 + 1 < 9x - 3 and graph the solutions on the real line."

### Approach #1 - Using "solve("

 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 "2x2 + 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 left-hand region, x = 1 from the middle region, and x = 5 from the right-hand region. The calculation "2x2 + 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.

### Approach #2 - Creating a User-Defined Function

The easy way to solve problems such as this on the TI-89 is to create a user-defined 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 TI-89:

 After starting a new problem, press 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> "solvepi" 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 "(" "f,x,", "s)" Finally, the function's body needs to be entered. First, type "=", then press <1> to select "solve(". To get the "sign(" function, you can select the math menu by pressing <2nd> , then press <1> to select the Number submenu, then press <8> to select "sign(", or: You can press , then "S", and use the arrow keys to scroll down to "sign(", or you can type the function by pressing <2nd> "sign" "(". You can get the "factor(" function by pressing <2>, or by typing it. To finish the body of the function, type "f,x))=" "s,x)" and press
To use the function to solve an inequality, follow these steps:
 To solve the inequality "2x2 + 1 < 9x - 3", you can either select the function name from the VAR-LINK menu (press <2nd>, then "s"), or type it by pressing <2nd> "solvepi" "(". The function takes three parameters: the function (Note that you need to move everything on the right side of the inequality over to the left. So this problem becomes "2x2 + 1 - 9x + 3") followed by a comma, then the variable ("x" in this case) followed by a comma, then either "1" for "f > 0", or "-1" for "f < 0".

### FAQ:

I know that the third parameter in the "solvepi" function is 1 for ">" and -1 for "<", but what do I do if I need ">=" or "<="?

If you need ">=", you can use "solvepi" to solve the ">" part, and "solve" (<F2><1>) to solve the "=" part.

How does the "solvepi" function work?

The function works, in general, by turning an inequality into an equation using the "sign" function. The function sign(x) has the value 1 if x > 0, and the value -1 if x < 0. The factor function forces the calculator to try to factor the polynomial, so that it will give its answer in terms of x, rather than some power of x (if possible).

## Rational Inequalities

In our particular course, we don't generally solve rational inequalities, but the method described by Tian leads to the following user-defined 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.

last update November 26, 2006 by JL Stanbrough