Solving Rational Inequalities and use sign diagram to sketch the graph. Image attached for better understanding. (3x^(2)+2x-1 )/(x+2) \ \textgreater \ 0

Question

Solving Rational Inequalities and use sign diagram to sketch the graph. Image attached for better understanding.


`(3x^(2)+2x-1 )/(x+2) \ \textgreater \ 0`

Answer 1

x ∈ (-2, -1) ∪ (1/3, ∞)

Explanation:

To solve this problem we must factor the expression that is shown in the Numerator of the inequality.

So, we have:

`3x^2 +2x -1 = 0\\3(x ^ 2+(2)/(3)x -(1)/(3)) = 0`

`3(x ^ 2+(2)/(3)x -(1)/(3)) = 0\\`

We should look for two numbers that add 2/3 as a result and multiply as a result -1/3

These numbers are -1/3 and 1

Then:

`3(x ^ 2+(2)/(3)x -(1)/(3)) = 3(x-(1)/(3))(x+1)`

So the roots are:

`x = -(1)/(3)\\\\x = 1`

Now the expression is as follows:

`(3(x-(1)/(3))(x+1))/((x + 2))>0`

Now we use the study of signs to solve this inequality.

We have 3 roots for the polynomials that compose the expression:

`x = -(1)/(3)\\\\x = 1\\\\x=-2`

We know that x = -2 is not in the domain of the function because it makes the denominator equal to zero

With these roots we make the study of signs:

Observe the attached image.

Note that:

`(x-(1)/(3))>0` when
`x>(1)/(3)`

`(x + 2)>0` when
`x>-2`

`(x + 1)>0` when
`x>-1`

Finally after the study of signs we can reach the conclusion that:

x ∈ (-2, -1) ∪ (1/3, ∞)