Question

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

$\frac{3x^{2}+2x-1 }{x+2} \ \textgreater \ 0$

Answer 1

Answer:

x ∈ (-2, -1) ∪ (1/3, ∞)

Step-by-step explanation:

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

So, we have:

$3x^2 +2x -1 = 0\\3(x ^ 2+\frac{2}{3}x -\frac{1}{3}) = 0$

$3(x ^ 2+\frac{2}{3}x -\frac{1}{3}) = 0$

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:

$3(x ^ 2+\frac{2}{3}x -\frac{1}{3}) = 3(x-\frac{1}{3})(x+1)$

So the roots are:

$x = -\frac{1}{3}\\\\x = 1$

Now the expression is as follows:

$\frac{3(x-\frac{1}{3})(x+1)}{(x + 2)}>0$

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

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

$x = -\frac{1}{3}\\\\x = 1\\\\x=-2$

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:

$(x-\frac{1}{3})>0$ when $x>\frac{1}{3}$

$(x + 2)>0$ 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, ∞)