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| | ! Step 1: | | ! Step 1: |
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| − | |First we replace the inequalities with equality. So <math>y = \vert x\vert + 1</math>, and <math>x^2 + y^2 = 9</math>. | + | |First we replace the inequalities with equality. So <math>y = 2x - 3</math>, and <math>y = 4 - x^2</math>. |
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| | |Now we graph both functions. | | |Now we graph both functions. |
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| | |point satisfies the inequality or not. For both equations we will pick the origin. | | |point satisfies the inequality or not. For both equations we will pick the origin. |
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| − | |<math>y < \vert x\vert + 1:</math> Plugging in the origin we get, <math>0 < \vert 0\vert + 1 = 1</math>. Since the inequality is satisfied shade the side of | + | |<math>y > 2x - 3:</math> Plugging in the origin we get, <math> 0 > 2(0) - 3 = -3 </math>. Since the inequality is false, we shade the side of |
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| − | |<math>y < \vert x\vert + 1</math> that includes the origin. We make the graph of <math>y < \vert x\vert + 1</math>, since the inequality is strict. | + | |<math>y = 2x - 3</math> that does not include the origin. We make the graph of <math>y < \vert x\vert + 1</math> dashed, since the inequality is strict. |
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| − | |<math>x^2 + y^2 \le 9:</math> <math>(0)^2 +(0)^2 = 0 \le 9</math>. Once again the inequality is satisfied. So we shade the inside of the circle. | + | |<math>y \le 4 - x^2:</math> Plugging in the origin we get <math>0 \le 4 - (0)^2 = 4</math>. Since this inequality is true, we shade the side of <math>y = 4 - x^2</math> that includes the origin. Here we make the graph of <math> y = 4 - x^2 </math> solid since the inequality sign is <math>\le</math> |
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| − | |We also shade the boundary of the circle since the inequality is <math>\le</math>
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| | |} | | |} |
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| | ! Final Answer: | | ! Final Answer: |
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| − | |The final solution is the portion of the graph that below <math>y = \vert x\vert + 1</math> and inside <math> x^2 + y^2 = 9</math> | + | |The final solution is the portion of the graph that below <math>y = 4 - x^2</math> and above <math> y = 2x - 3</math> |
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| | |The region we are referring to is shaded both blue and red. | | |The region we are referring to is shaded both blue and red. |
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| − | |[[File:8A_Final_5.png]] | + | |[[File:4_Sample_Final_4.png]] |
| | |} | | |} |
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| | [[004 Sample Final A|<u>'''Return to Sample Exam</u>''']] | | [[004 Sample Final A|<u>'''Return to Sample Exam</u>''']] |
Latest revision as of 09:08, 2 June 2015
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 > 2x - 3 \qquad y \le 4-x^2}
Solution:
| ExpandStep 1:
|
| First we replace the inequalities with equality. So 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 = 2x - 3}
, and 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 = 4 - x^2}
.
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| Now we graph both functions.
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| ExpandStep 2:
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| Now that we have graphed both functions we need to know which region to shade with respect to each graph.
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| To do this we pick a point an equation and a point not on the graph of that equation. We then check if the
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| point satisfies the inequality or not. For both equations we will pick the origin.
|
| 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 > 2x - 3:}
Plugging in the origin we get, 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 0 > 2(0) - 3 = -3 }
. Since the inequality is false, we shade the side of
|
 that does not include the origin. We make the graph of 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}
dashed, since the inequality is strict.
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| 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 \le 4 - x^2:}
Plugging in the origin we get 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 0 \le 4 - (0)^2 = 4}
. Since this inequality is true, we shade the side of 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 = 4 - x^2}
that includes the origin. Here we make the graph of 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 = 4 - x^2 }
solid since the inequality sign is 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 \le}
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| ExpandFinal Answer:
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| The final solution is the portion of the graph that below 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 = 4 - x^2}
and above 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 = 2x - 3}
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| The region we are referring to is shaded both blue and red.
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|
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