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Graphing the Intersection of Regions>
Graphing the Intersection of Regions 2Graphing the Intersection of Regions 2
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Question 1 of 4
1. Question
Graph the intersection of regions`x^2+y^2≤4`$$|x|>1$$Hint
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Remember the following notations when graphing inequalities.Symbol Solid / Dotted `<` Dotted Line `>` Dotted Line `≤` Solid Line `≥` Solid Line First, treat the inequality sign as an equals sign and plot the curve.Graph the line `x=1` and `x=-1`, which are the absolute values of `x`
Since `x``>``1` and `x``<``-1`, we can determine that the shaded part would be the left side from `x=-1` and the right side from `x=1`
Next, graph the curve `x^2+y^2=4`
Use a test point to see which side of the line is to be shaded. We can try the origin, `(``0``,``0``)`$$\color{#9a00c7}{x}^2+\color{#00880A}{y}^2$$ `≤` `4` $$\color{#9a00c7}{0}^2+\color{#00880A}{0}^2$$ `≤` `4` Substitute `(0,0)` `0` `≤` `4` The inequality is true, so we will shade the side of the line which includes the origin
Finally, shade the overlapping regions.
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Question 2 of 4
2. Question
Graph the intersection of regions`y≥x^2``y≤3x`$$x<1$$Hint
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Remember the following notations when graphing inequalities.Symbol Solid / Dotted `<` Dotted Line `>` Dotted Line `≤` Solid Line `≥` Solid Line First, treat the inequality sign as an equals sign and plot the curve.Graph the line `x=1`
Since `x<1`, we can determine that the shaded part would be the left side because all values on the left side are less than `1`
Next, graph the line `y=3x`
Use a test point to see which side of the line is to be shaded. We can try the point `(``2``,``0``)``y` `≤` `3``x` `0` `≤` `3(``2``)` Substitute `(2,0)` `0` `≤` `6` The inequality is true, so we will shade the side of the line which includes the point `(2,0)`
Next, graph the curve `y=x^2`
Use a test point to see which side of the line is to be shaded. We can try the point `(``0``,``2``)``y` `≥` $$\color{#9a00c7}{x}^2$$ `2` `≥` $$\color{#9a00c7}{0}^2$$ Substitute `(0,2)` `2` `≥` `0` The inequality is true, so we will shade the side of the line which includes the point `(0,2)`
Finally, shade the overlapping regions.
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Question 3 of 4
3. Question
Graph the intersection of regions`y≤x^2``x≤2`$$y>-1$$Hint
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Remember the following notations when graphing inequalities.Symbol Solid / Dotted `<` Dotted Line `>` Dotted Line `≤` Solid Line `≥` Solid Line First, treat the inequality sign as an equals sign and plot the curve.Graph the line `y=-1`
Since `y> -1`, we can determine that the shaded part would be the upper side because all values on the upper side are greater than `-1`
Next, graph the line `x=2`
Since `x≤2`, we can determine that the shaded part would be the left side because all values on the left side are less than `2`
Next, graph the curve `y=x^2`
Use a test point to see which side of the line is to be shaded. We can try the point `(``0``,``3``)``y` `≤` $$\color{#9a00c7}{x}^2$$ `3` `≤` $$\color{#9a00c7}{0}^2$$ Substitute `(0,2)` `3` `≤` `0` The inequality is false, so we will shade the side of the line which does not include the point `(0,3)`
Finally, shade the overlapping regions.
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Question 4 of 4
4. Question
Graph the intersection of regions`x^2+y^2≤4`$$y>1$$$$x+y>2$$Hint
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Remember the following notations when graphing inequalities.Symbol Solid / Dotted `<` Dotted Line `>` Dotted Line `≤` Solid Line `≥` Solid Line First, treat the inequality sign as an equals sign and plot the curve.Graph the line `x+y=2`
Use a test point to see which side of the line is to be shaded. We can try the origin, `(``0``,``0``)`$$\color{#9a00c7}{x}+\color{#00880A}{y}$$ `>` `2` $$\color{#9a00c7}{0}+\color{#00880A}{0}$$ `>` `2` Substitute `(0,0)` `0` `>` `2` The inequality is false, so we will shade the side of the line which does not include the origin
Next, graph the line `y=1`
Since `y>1`, we can determine that the shaded part would be the upper side because all values on the upper side are greater than `1`
Next, graph the curve `x^2+y^2=4`
Use a test point to see which side of the line is to be shaded. We can try the origin, `(``0``,``0``)`$$\color{#9a00c7}{x}^2+\color{#00880A}{y}^2$$ `≤` `4` $$\color{#9a00c7}{0}^2+\color{#00880A}{0}^2$$ `≤` `4` Substitute `(0,0)` `0` `≤` `4` The inequality is true, so we will shade the side of the line which includes the origin
Finally, shade the overlapping regions.