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 $(1, 0)$.