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).)
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."
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. |
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Next, enter the related equation "2x2 + 1 = 9x - 3" and solve it for x. |
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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. |
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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". |
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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> |
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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)" |
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Finally, the function's body needs to be entered. First, type "=", then press <F2><1> to select "solve(". |
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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: |
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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> "(". |
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You can get the "factor(" function by pressing <F2> <2>, or by typing it. |
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To finish the body of the function, type <alpha> "f,x))=" <alpha> "s,x)" and press <ENTER> |
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> "(". |
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The function takes three parameters:
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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 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.