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Solving Polynomial Inequalities

on the TI-89


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.

The Algebra Menu

Next, enter the related equation "2x2 + 1 = 9x - 3" and solve it for x.

solve(2x^2 + 1 = 9x - 3, 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.

Number Line

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.

Checking the result

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 <F4> and select "1: Define".

f4 - 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>

Entering the function name

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)"

Entering function parameters

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

Entering "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:

The Math/Num menu

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> "(".

The Catalog screen

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

The "factor(" function

To finish the body of the function, type <alpha> "f,x))=" <alpha> "s,x)" and press <ENTER>

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><VAR-LINK>, then "s"), or type it by pressing <2nd><alpha> "solvepi" <alpha> "(".

The Var-Link Menu

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".

Solving the inequality


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