An Inequality in a Book - Sự bất bình đẳng trong một cuốn sách

You can read the book here

You can download it directly from here.
                                 
             Usually the first problem in a book is easier. Look at it

An algebraic solution I did not find despite the obvious geometric meaning.
Here is the picture with the solution in the book.

True, she is beautiful............

          
            We present below the geometrical meaning of the problem. 
    
           The equation 

(1)                                        $9x^{2}+8xy+7y^{2}=6$

 represent an ellipse; thanks to $GEOGEBRA$ she looks like this

You can see in the image above the point $A(\frac{1}{2};\frac{1}{2})$ that belongs to the ellipse (drawn a little clumsy). The center of this ellipse still remain origin $O(0;0)$ but the axes are the lines $y=\frac{\sqrt{17}-1}{4}x$ , $y=-\frac{\sqrt{17}+1}{4}x$ as will be seen in a later image.
         
         The inequality given in the statement $9a^{2}+8ab+7b^{2} \leqslant 6$ says about the point $M(a:b)$ it is inside the ellipse (1).

          The equation

(2)                                             $7x+5y+12xy=9$

reprezent a hyperbola with asymptotes $y=-\frac{7}{12}$ and $x=-\frac{5}{12}$, as shown below and point $A(\frac{1}{2};\frac{1}{2})$ also belongs to her.

           It is important that the point $A(\frac{1}{2};\frac{1}{2})$ belongs to both conics but crucial is that these conics are tangent at $A$  (and both at line $13x+11y=12$); see the image below

         Now, inequality $7a+5b+12ab \leqslant 9$ says that point $M(a;b)$ is outer the branches of hyperbola which is (geometrically) obvious.

$ \blacksquare$

         I hope that through these we have dismantled the mechanism that made this kind of problems to appear.
                                                      GOOD  LOCK !