Simple Inequalities

$x^{400}<4^{600}$ or $x<4^{\frac{600}{400}}=4^{\frac{3}{2}}=(4^{\frac{1}{2}})^{3}= 8$ . This is if $x$ is positive. If $x$ is negative, the inequality sign is flipped i.e. $x>-8$

Since we are looking for the greatest value of $x$ , we take take the positive value of $x$ i.e. the condition $x<8$ Hence the greatest integral value of $x$ is $7$ .