For the function g(x)= x-2 / x+4 , solve g(x) < 0 . Please make sure you show the work

QUESTION: g(x)=(x-2)/(x+4) ; solve g(x)<0

SOLUTION:

The numerator of this ratio is (x-2); the denumerator is (x+4).

Let start solving separately the two simple inequalities (x-2)>0 and (x+4)>0.

Numerator : (x-2)>0 means x>2;

Denominator: (x+4)>0 means x>-4.

So, in this way we know that the numerator is positive when x is greater than 2 while the denominator is positive when x is greater than -4.

Asking to solve g(x) <0 means that the quotient has to be negative. That could be only if "the numerator is positive and the denominator is negative" or viceversa "the numerator is negative and the denominator positive".

Let me use a graph:

-4 2

------|----------------|-------------- real line

Num. - - +

Den. - + +

Is it clear now that the solution is the set of number between -4and 2, that is the open interval (-4;2).

Numerator : (x-2)>0 means x>2;

Denominator: (x+4)>0 means x>-4.

So, in this way we know that the numerator is positive when x is greater than 2 while the denominator is positive when x is greater than -4.

Asking to solve g(x) <0 means that the quotient has to be negative. That could be only if "the numerator is positive and the denominator is negative" or viceversa "the numerator is negative and the denominator positive".

Let me use a graph:

-4 2

------|----------------|-------------- real line

Num. - - +

Den. - + +

Is it clear now that the solution is the set of number between -4and 2, that is the open interval (-4;2).