Answer
Please see the graph.
Work Step by Step
Green lines: $x^{2}-y^{2}\ge1$
Purple line: $x\ge0$
$x^{2}-y^{2}\ge1$
This is the equation of a hyperbola. We have three regions to test to determine what region(s) to shade: $(-∞, -1]$, $[-1, 1]$, $[1, ∞)$.
We pick the following values of $x$ to test (with keeping the $y$ coordinate $0$): $-2$, $0$, and $2$.
$(-2,0)$
$x^{2}-y^{2}\ge1$
$(-2)^{2}-0^{2}\ge1$
$4 - 0 \ge 1$
$4 \ge 1$ (true, so we shade this region)
$(0,0)$
$x^{2}-y^{2}\ge1$
$0^{2}-0^{2}\ge1$
$0 - 0 \ge 1$
$0 \ge 1$ (false)
$(2,0)$
$x^{2}-y^{2}\ge1$
$(2)^{2}-0^{2}\ge1$
$4 - 0 \ge 1$
$4 \ge 1$ (true, so we shade this region)
$x\ge0$
We pick the point $(1,1)$ to determine what region to shade.
$(1,1)$
$x\ge0$
$1\ge0$ (true, so we shade this region)
The solution set to the inequalities is the overlap of the two graphs.
