# 004 Sample Final A, Problem 4

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Graph the system of inequalities. ${\displaystyle y>2x-3\qquad y\leq 4-x^{2}}$ Solution:

Step 1:
First we replace the inequalities with equality. So ${\displaystyle y=2x-3}$, and ${\displaystyle y=4-x^{2}}$.
Now we graph both functions.
Step 2:
Now that we have graphed both functions we need to know which region to shade with respect to each graph.
To do this we pick a point an equation and a point not on the graph of that equation. We then check if the
point satisfies the inequality or not. For both equations we will pick the origin.
${\displaystyle y>2x-3:}$ Plugging in the origin we get, ${\displaystyle 0>2(0)-3=-3}$. Since the inequality is false, we shade the side of
${\displaystyle y=2x-3}$ that does not include the origin. We make the graph of ${\displaystyle y<\vert x\vert +1}$ dashed, since the inequality is strict.
${\displaystyle y\leq 4-x^{2}:}$ Plugging in the origin we get ${\displaystyle 0\leq 4-(0)^{2}=4}$. Since this inequality is true, we shade the side of ${\displaystyle y=4-x^{2}}$ that includes the origin. Here we make the graph of ${\displaystyle y=4-x^{2}}$ solid since the inequality sign is ${\displaystyle \leq }$
The final solution is the portion of the graph that below ${\displaystyle y=4-x^{2}}$ and above ${\displaystyle y=2x-3}$