Exercise 30 Page 643

We want to solve the given system of inequalities by graphing. Let's graph each inequality one at a time.

Inequality (I)

We can write the boundary curve for Inequality (I) by replacing the less than or equal to sign with an equals sign. The above equation is an equation of hyperbola. Let's write it in a standard form. We are given the equation of the vertical hyperbola. Let's find necessary information to draw its graph.

Vertical Hyperbola with Center $(0,0)$
Equation	$\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$
Transverse axis	Vertical
Vertices	$(0,\pm a)$
Foci	$(0,\pm c),$ where $c^2=a^2+b^2$
Asymptotes	$y=\pm \frac{a}{b}x$

Using this information, we can identify that the vertices are $\left(0,\pm \sqrt{3}\right).$ Let's substitute $a=\sqrt{3}$ and $b=2\sqrt{2}$ into the formula for the asymptotes and obtain their equations.