SOLUTION: solve and graph {{{ 6/((x+4))+1<=4/((3x+12)) }}}.

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Question 1102656: solve and graph
+6%2F%28%28x%2B4%29%29%2B1%3C=4%2F%28%283x%2B12%29%29+.
Found 3 solutions by greenestamps, josgarithmetic, ikleyn:
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


Get all terms on one side of the inequality (with 0 on the other side), and with a common denominator.

+6%2F%28%28x%2B4%29%29%2B1%3C=4%2F%28%283x%2B12%29%29+
18%2F%283x%2B12%29%2B%283x%2B12%29%2F%283x%2B12%29-4%2F%283x%2B12%29%3C=0
%283x%2B26%29%2F%283x%2B12%29%3C=0
%28x%2B26%2F3%29%2F%28x%2B4%29%3C=+0

The critical points -- where numerator or denominator are 0, and therefore the only places where the function value can change sign -- are -26/3 and -4.

Choosing test points (or any other valid method) will show that the function value is positive to the left of -26/3 and to the right of -4; it is negative only between -26/3 and -4.

If you are graphing the solution set on a number line, the left endpoint -26/3 is included in the solution set because the numerator can be 0; the right endpoint is not included because the denominator cannot be 0.

So the solution set for the inequality in interval notation is [-26/3,-4).

The solution set of the inequality can be seen in a graph of the equation:

graph%28400%2C200%2C-30%2C10%2C-10%2C10%2C%28x%2B26%2F3%29%2F%28x%2B4%29%29

Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
This is only a start - not the whole solution.

Recognize that whatever you find, x%3C%3E-4.

6%2F%28x%2B4%29-4%2F%283x%2B12%29%2B1%3C=0
Common denominator 3%28x%2B4%29;

18%2F%283x%2B12%29-4%2F%283x%2B12%29%2B%283x%2B12%29%2F%283x%2B12%29%3C=0


---------------Below here is faulty because accounting for both numerator AND denominator and critical x values must be done------------------------------
18-4%2B3x%2B12%3C=0
3x%2B12%2B18-4%3C=0
3x%2B26%3C=0
recheck for any mistakes, and continue...

Answer by ikleyn(52780) About Me  (Show Source):
You can put this solution on YOUR website!
.
Approach used and shown by by @josgarithmetic is   W R O N G.

A V O I D   it.


In the solution, you must analyse and take into account the signs of BOTH the numerator and denominator, as @greenestamps did.


Also see the lesson
    - Solving inequalities for rational functions with non-zero right side
in this site.