inequalities

[fiery123]

solve 1/(5-4x) + 1/(x-7) <= 0

[eranga262154]

It is depend on the basis of inequalities...

First simplify the expression as follows.

(-3x - 2) /{(5 - 4x)(x - 7)} <= 0

To satisfied the above condition,

(-3x - 2) <= 0

and (5 - 4x) and (x - 7) not equal to zero.

So,

x >= 2/3 and

not equal 5/4 and 7.

[Logophobic]

That solution is clearly not correct. Just let x = 2/3, then calculate the expression: 0.27067669, which is not less than or equal to zero.

The correct solution is:
x <= -2/3; 5/4 < x < 7

I found this by graphing the expression. I'm not sure how to properly arrive at this solution, except the first part may be:

1/(5-4x) + 1/(x-7) <= 0
1/(5-4x) <= 1/(7-x)
5 - 4x >= 7 - x
-3x >= 2
-x >= 2/3
x <= -2/3

[eranga262154]

Can you put your graph here.Can you say where I'm going wrong.