Solve (3x+ 1/ x + 4 ) ≥ 1.
Solution
The first thing that we have to do here is subtract 1 from both of sides and then get everything in a single rational expression.
(3x + 1/ x + 4) -1 ≥ 0
(3x + 1/x+4) - (x + 4/x+4) ≥ 0
(3x +1 - ( x + 4))/ x + 4 ≥ 0
2 x - 3 / x + 4 ≥ 0
In this case there is no factoring to do thus we can go straight to recognizing where the numerator & denominator are zero.
numerator : x = 3/2 denominator : x = -4
Following is the number line for this problem.
Okay, we desire values of x that give positive and/or zero in the rational expression. It looks like the outer two regions as well as x = 3/2 . As with the first instance we will have to avoid x = -4 since that will give a division by zero error.
Then the solution for this problem is,
-∞ < x < -4 and 3/2 ≤ x < ∞
( -∞, -4) and [3 , ∞ )