Solving Inequalities: How Do I Determine the Correct Answer?

[member 529879]

How would I solve the inequality (X-4)/X>0. I thought that inequalities were solved in the same way equations were, but when I solve that way I get X>4 which isn't the entire answer.

 

[RUber]

For this one, it is best to look at critical points where either the top or bottom equal zero. From that, you should be able to quickly categorize the intervals where the expression is true.

 

[Mark44]

How would I solve the inequality (X-4)/X>0. I thought that inequalities were solved in the same way equations were

No, they're not. For example, if you multiply both sides of an equation by, say, -1, you get a new equation that is equivalent to the one you started with.

If you multiply an inequality by -1, the inequality symbol changes direction.

, but when I solve that way I get X>4 which isn't the entire answer.

 

[Svein]

Well, what you do is:
x-4: Negative when x<4, positive when x>4
x: Negative when x<0, positive when x>0
Expression: Positive when x<0 (neg. and neg. makes pos.), negative when 0<x<4 and positive when x>4

 

[Mark44]

Well, what you do is:
x-4: Negative when x<4, positive when x>4
x: Negative when x<0, positive when x>0
Expression: Positive when x<0 (neg. and neg. makes pos.), negative when 0<x<4 and positive when x>4

Or, more simply, just multiply both sides of the inequality by x², first making a note that x cannot be zero. For x ≠ 0, x² > 0, so the direction of the inequality doesn't change.

 

[member 529879]

When I solve that way I get x>0 and x>4. Am I doing something wrong?

 

[RUber]

The correct answer was given in post 4.
$\frac{x-4}{x} >0$
Taking Mark44's recommendation, this could also be seen as:
$x^2\frac{x-4}{x} >0*x^2$
$x(x-4) >0$
Remember that (-)(-)=(+) and (+)(+)=(+), and (-)(+)=(-) just the same as (-)/(-)=(+) and (+)/(+)=(+), and (-)/(+)=(-).
So whether or not you multiply by $x^2$, you still need to find the signs of your terms (x-4) and (x) and the appropriate regions.
Build a simple table, the inequality will only hold true if both terms are negative or both are positive.