College Algebra, Chapter 2, 2.2, Section 2.2, Problem 90

Sketch the region $\{(x, y) | 2x < x^2 + y^2 \leq 4\}$

Let's simplify the inequality

$2x < x^2 + y^2 $ and $ x^2 + y^2 \leq 4$

We have,

$x^2 + y^2 \leq 4 \qquad$ The equation of all circles with radius $\leq 2$ and center at $(0, 0)$.

and


$
\begin{equation}
\begin{aligned}

& 2x < x^2 + y^2
&& \text{Model}
\\
\\
& 0 < -2x + x^2 + y^2
&& \text{Subtract } 2x
\\
\\
& 1 < (x^2 - 2x + 1) + y^2
&& \text{Complete the square: add } \left( \frac{-2}{2} \right)^2 = 1
\\
\\
& 1 < (x - 1)^2 + y^2
&& \text{Perfect Square}
\\
\\
& 1 < (x - 1)^2 + y^2
&& \text{The equation of all circles with radius $> 1$ and centers at $(1, 0)$}

\end{aligned}
\end{equation}
$





So the shaded region is all the circles with radius $\leq 2$ and center at $(0,0)$ but greater than the circles with radius $> 1$ and center at $(1, 0)$.