Request for solving 2 inequalities

[Math Tutor]

I am writing to request for your help in solving the following mathematics problems:
1. solve the inequality:
[tex]x^3-x^2-5x+3>0[/tex]
2. solve the inequality:
[tex]f(x) =(x+3)/(x-4)<0[/tex]

thanks in advance as you help me with these problems.
Lazarus Mgbelu

[Guest]

Let us find the roots of the equation by hit and trial method

x^3 - x^2 -5x -3
put x = 0, 1, -1

at x = 0
=> 0 - 0 - 0 - 3 is not equal to 0

at x = 1
=> 1 - 1 - 5 - 3 is not equal to 0

at x = -1
=> -1 - 1 + 5 - 3 = 0
=> x = -1 is the root of equation

For other roots
put x^2 - 2x - 3 = 0
=> x^2 - 3x + x - 3 = 0
=> x(x - 3) + (x - 3) = 0
=> x= -1 , 3

Given inequality reduced as
(x - 1)(x + 1)(x - 3) > 0
=> x - 1 > 0 or (x + 1) > 0 or (x - 3) > 0
(By Zero product property)

=> x > 0 or x > -1 or x > 3


2) f(x) = (x + 3) / (x - 4) < 0
Multiply each side by (x - 4)

(x - 4) * (x + 3) / (x - 4) < 0* (x - 4)
x + 3 < 0

or x < -3