2x (6+1/2x)< 2/3x+1.3x

Question

2x(6+1/2)<2/3+1.3x

Answer 1

(1) 2x(6+½x)<⅔+1.3x,

12x+x²<⅔+1.3x, multiply through by 30:

360x+30x²<20+39x,

30x²+321x-20<0. Using the quadratic formula:

x=(-321±√(103041+2400))/60=(-321±√105441)/60=(-321±324.7168)/60,

x=0.06195 or -10.76195.

Now we have to deal with the inequality.

We can write the quadratic inequality: (x-0.06195)(x+10.76195)<0.

To get a negative product (a) x-0.06195<0 and x+10.76195>0; or (b) x-0.06195>0 and x+10.76195<0.

(a) x<0.06195 and x>-10.76195, that is, -10.76195<x<0.06195 (x is between two limits).

(b) cannot be satisfied. ANSWER: -10.76195<x<0.06195.

(2) 2x(6+½x)<⅔x+1.3x,

12x+x²<⅔x+1.3x, multiply through by 30:

360x+30x²<20x+39x,

301x+30x²<0,

x(301+30x)<0.

So x<0 and 301+30x>0, 30x>-301, x>-301/30, that is, -301/30<x<0;

or x>0 and 30x<-301 which cannot be satisfied. ANSWER -301/30<x<0.