Graph the solution set of each system of inequalities.
y ≥ (x - 2)² + 3
y ≤ -(x - 1)² + 6

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Identify the two inequalities given: $y \geq (x - 3)^2 - 5$ and $y \leq -(x - 4)^2 + 3$.

Recognize that each inequality represents a region bounded by a parabola. The first parabola opens upwards with vertex at $(3, -5)$, and the second parabola opens downwards with vertex at $(4, 3)$.

Graph the parabola $y = (x - 3)^2 - 5$ as a solid curve because the inequality includes equality ($\geq$). Shade the region above this parabola since $y$ is greater than or equal to the parabola.

Graph the parabola $y = -(x - 4)^2 + 3$ as a solid curve because the inequality includes equality ($\leq$). Shade the region below this parabola since $y$ is less than or equal to the parabola.

The solution set to the system is the intersection of the two shaded regions. This means the area where the shaded region above the first parabola overlaps with the shaded region below the second parabola.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Graphing Quadratic Inequalities

Graphing quadratic inequalities involves plotting the parabola defined by the quadratic equation and then shading the region that satisfies the inequality. For 'y ≥ f(x)', shade above the parabola, and for 'y ≤ f(x)', shade below. The boundary parabola is included if the inequality is '≥' or '≤'.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is y = a(x - h)² + k, where (h, k) is the vertex of the parabola. This form makes it easy to identify the parabola's vertex and direction (upward if a > 0, downward if a < 0), which is essential for graphing and understanding the inequality regions.

Solution Set of a System of Inequalities

The solution set of a system of inequalities is the region where the shaded areas of all inequalities overlap. It represents all points that satisfy every inequality simultaneously. Identifying this intersection is key to solving and graphing systems of inequalities.

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Textbook Question

Find the partial fraction decomposition for each rational expression. 5-2x / (x² + 2)(x - 1)

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Textbook Question

Let $A = [\begin{matrix} - 2 & 4 \\ 0 & 3 \end{matrix}]$ and $B = $\left$[ $\begin{matrix}$ -6 & 2 \\ 4 & 0 $\end{matrix}$ $\right$]$ . Find each of the following.