Inequality My TI89 does not solve the inequality (3x + 1)/x < 0. However several students of mine can. How is this possible?

There is nothing to solve because you do not have an equation, nor an inequation (inequality). You need an equal sign, >,<, etc.
All you can do with this polynomial is try to factor it. Here is the factored out form
x^3-3x^2-4x=x(x^2-3x-4)=x(x-4)(x+1)
To factor (x^2-3x-4) look for the divisors of -4 whose sum is -3
You cannot do that on your calculator because is does not have a Computer Algebra System or CAS.

Sorry, the calculator cannot solve such ualities (nothing to do with the absolute function).
To find the solution set you need to study the two cases for the absolute value.
Case 1
If x-1> 0, the absolute value is x-1. The inequality takes the form x-1<9. You can solve it as x<10

Case 2
If x-1<0, then the absolute value is 1-x. The inequality becomes 1-x<9. The solution for this inequality is x>-8

Combine the two domains to find the intersection
-8<x<10

Solve the two sided inequality and show the solution on real line

7 < 1-2x ? 10

Solve the two sided inequality and show the solution on real line

7 < 1-2x ≤ 10

And your question is? If you just want to know how to solve this system, you only need to substitute one equation into the other and solve for the remaining unknown.

y=3x (equation 2), so plug it into equation 1

7x-2y=1
7x-2(3x)=1
7x-6x=1
x=1

from equation 2 then, you know y=3x=3

these calculators have a small ram memory battery inside open the case remove it then put it back in "think watch battery" this happened to mine.

This book is what I used to help learn how to solve mesh circuits. A high education level helps. You just set it up as you see it in a simultaneous equation.

Electrical Engineering Applications with the TI-89
Electrical Engineering students will find the 70 topics covered in this book provide in-depth analyses of how to solve many common engineering problems using the powerful CAS system of the TI-89.
Author(s) - David R. Voltmer and Mark A. Yoder
[email protected]

The possible solutions are
x= -y/3
y= -3x/1

See captured image below

Just plug in one side of the equation and find the answer. Then plug in the other side and compare answers.