Answer
Please see the graph.
Work Step by Step
Orange line: $x^{2}+y^{2}>1$
Red line: $\frac{x^{2}}{4}-y^{2}\ge1$
The orange line and red line will be a dotted line and a solid line, respectively, since the two inequalities (respectively) have a greater than sign and a greater than or equal to sign.
The orange line is the equation of a circle with radius 1. If a point is outside the circle, it is part of the solution set.
The red line is the equation of a hyperbola. For the hyperbola, we pick three $x$ values (keeping the $y$ value the same) to determine what region(s) to shade.
Hyperbola: $(-3,0)$, $(0,0)$, $(3,0)$
$(-3,0)$
$\frac{x^{2}}{4}-y^{2}\ge1$
$\frac{(-3)^{2}}{4}-0^{2}\ge1$
$\frac{9}{4}-0\ge1$
$9/4 \ge 1$ (true, so we shade this region)
$(0,0)$
$\frac{x^{2}}{4}-y^{2}\ge1$
$\frac{0^{2}}{4}-0^{2}\ge1$
$\frac{0}{4}-0\ge1$
$0-0 \ge 1$
$0 \ge 1$ (false)
$(3,0)$
$\frac{x^{2}}{4}-y^{2}\ge1$
$\frac{(3)^{2}}{4}-0^{2}\ge1$
$\frac{9}{4}-0\ge1$
$9/4 \ge 1$ (true, so we shade this region)
$(0,0)$
$x^{2}+y^{2}>1$
$0^{2}+0^{2}>1$
$0 + 0 > 1$
$0 > 1$ (false, so we shade outside the circle)
The overlap of the two graphs is the solution set to the system of inequalities.
