How do you solve (x+3)/(3x-6)<=2?

2 Answers
May 9, 2018

x in (-oo, 2) uu [3, oo)

Explanation:

As per the question, we have

(x+3)/(3x-6) <= 2

:. (x+3)/(3x-6) - 2 <= 0

:. (x + 3 -6x + 12)/(3x - 6) <= 0

:. (-5x + 15)/(3x - 6) <= 0

:. (5x - 15)/(3x - 6) >= 0

:. (5(x-3))/(3(x-2)) >= 0

:. By Wavy Curve Method we get,

MS PaintMS Paint
Note : In the image the orange region represents the area on the numberline to which x belongs, i.e from -oo to 2 and 3 to oo.

x in (-oo, 2) uu [3, oo)

Hence, the answer.

May 9, 2018

color(blue)((-oo,2)uu[3,oo)

Explanation:

(x+3)/(3x-6) <=2

Multiply by (3x-6)^2 ( valid since this is always positive )

(3x-6)(x+3)<=2(3x-6)^2

3x^2+3x-18<=18x^2-72x+72

15x^2-75x+90>=0

Factor:

(15x-30)(x-3)>=0

(15x-30)(x-3)=0=>x=2,x=3

Using signs of brackets:

For:

2<=x<=3

+ - >=0color(white)(888) False

For:

x<=2

- - >=0color(white)(888) True

For:

x>=3

+ + >=0color(white)(888) True

Solutions in interval notation:

(-oo,2)uu[3,oo)

Notice the use of a open interval for 2, this is because for x= 2 the denominator would be zero.