Question

1. (20 points) graph the system of inequalities. then find the coordinates of the points of intersection of the graphs of the related equations.

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$$\begin{cases} y \\le 3 - x^2 \\\\ y \\ge x + 1 \\end{cases}$$

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Explanation:

Step1: Set equations equal

$$3 - x^2 = x + 1$$

Step2: Rearrange into standard form

$$x^2 + x - 2 = 0$$

Step3: Factor the quadratic equation

$$(x + 2)(x - 1) = 0$$

Step4: Solve for x-coordinates

$$x = -2\text{ or }x = 1$$

Step5: Find corresponding y-coordinates

$$y = -2 + 1 = -1\text{ and }y = 1 + 1 = 2$$

Answer:

The coordinates of the points of intersection are:

$$(-2, -1)\text{ and }(1, 2)$$