008A Sample Final A, Question 5

Question: Graph the system of inequalities Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y < \vert x\vert +1 }           $x^{2}+y^{2}\leq 9$

Foundations
1) What do the graphs of $y=\vert x\vert +1$, and $x^{2}+y^{2}=9$ look like?
2) Each graph splits the plane into two regions. Which one do you want to shade?
Answer:
1) The first graph looks like a V with the vertex at (0, 1), the latter is a circle centered at the origin with radius 3.
2) Since the Y-value must be less than Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \vert x\vert +1} , shade below the V. For the circle shde the inside.

Solution:

Step 1:
First we replace the inequalities with equality. So $y=\vert x\vert +1$, and $x^{2}+y^{2}=9$.
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.
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y<\vert x\vert +1:} Plugging in the origin we get, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 0<\vert 0\vert +1=1} . Since the inequality is satisfied shade the side of
$y<\vert x\vert +1$ that includes the origin. We make the graph of $y<\vert x\vert +1$, since the inequality is strict.
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x^{2}+y^{2}\leq 9:} Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (0)^{2}+(0)^{2}=0\leq 9} . Once again the inequality is satisfied. So we shade the inside of the circle.
We also shade the boundary of the circle since the inequality is $\leq$

Step 3:
The final solution is the portion of the graph that below $y=\vert x\vert +1$ and inside $x^{2}+y^{2}=9$

Final Answer:
The final solution is the portion of the graph that below $y=\vert x\vert +1$ and inside Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2 + y^2 = 9}

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