7/x-3 + 2x/x+3 < -1

Question

Solving for values of x to graph on the number line. 

Answer 1

7/(x-3) + 2x/(x+3) < -1

 

7(x+3)/((x+3)(x-3)) + 2x(x-3)/((x+3)(x-3))  + 1 < 0

 

(7(x+3) + 2x(x-3))/(x^2-9) + (x^2-9)/(x^2-9) < 0

 

(7x + 21 + 2x^2 - 6x + x^2 - 9)/(x^2-9) < 0

 

(3x^2 + x + 12)/(x^2-9) < 0

 

h = -1/(2*3) = - 1/6

k = 12 - 3(-1/6)^2 = 12 - 1/12 > 0

 

3 > 0 and k > 0 ==> numerator always positive.

 

So denominator determines sign of fraction:

 

x^2 - 9 < 0

 

x^2 < 9

 

|x| < 3

 

-3 < x < 3 is solution.

 

In interval notation: x ∈ (–3,3).