Determining interval solution of irrational inequality

I have to solve irrational inequality:

$\displaystyle \sqrt x + \sqrt {x + 7} + 2\sqrt {x^2 + 7x} < 35 - 2x$

Inequality is defined for $\displaystyle 0 \le x < 17.5$

Transforming inequality we get that:

$\displaystyle 144x^2 - 2137x + 7569 > 0$

Solving this quadratic inequality we get that

$\displaystyle x_1 = 9 \wedge x_2 = \frac{{841}}{{144}} \approx 5.84$

So, solutions of beggining irrational inequality must be either

$\displaystyle 0 \le x < 5.84$ or $\displaystyle 9 < x < 17.5$.

In book stands that solution is $\displaystyle 0 \le x < 5.84$, but

how do I know that this interval is solution?

I mean, how do I determine exactly which interval is solution?