Graphing the Intersection of Regions 2

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Question 1 of 4

1. Question
Graph the intersection of regions

$x^{2} + y^{2} \leq 4$ $x^2+y^2≤4$

$|x|>1$
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Remember the following notations when graphing inequalities.

Symbol Solid / Dotted

$<$ $<$ Dotted Line

$>$ $>$ Dotted Line

$\leq$ $≤$ Solid Line

$\geq$ $≥$ Solid Line

First, treat the inequality sign as an equals sign and plot the curve.

Graph the line $x = 1$ $x=1$ and $x = - 1$ $x=-1$ , which are the absolute values of $x$ $x$

Since $x$ $x$ $>$ $>$ $1$ $1$ and $x$ $x$ $<$ $<$ $- 1$ $-1$ , we can determine that the shaded part would be the left side from $x = - 1$ $x=-1$ and the right side from $x = 1$ $x=1$

Next, graph the curve $x^{2} + y^{2} = 4$ $x^2+y^2=4$

Use a test point to see which side of the line is to be shaded. We can try the origin, $($ $($ $0$ $0$ $,$ $,$ $0$ $0$ $)$ $)$

$\color{#9a00c7}{x}^2+\color{#00880A}{y}^2$ $\leq$ $≤$ $4$ $4$

$\color{#9a00c7}{0}^2+\color{#00880A}{0}^2$ $\leq$ $≤$ $4$ $4$ Substitute $(0, 0)$ $(0,0)$

$0$ $0$ $\leq$ $≤$ $4$ $4$

The inequality is true, so we will shade the side of the line which includes the origin

Finally, shade the overlapping regions.
Question 2 of 4

2. Question
Graph the intersection of regions

$y \geq x^{2}$ $y≥x^2$

$y \leq 3 x$ $y≤3x$

$x<1$
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Remember the following notations when graphing inequalities.

Symbol Solid / Dotted

$<$ $<$ Dotted Line

$>$ $>$ Dotted Line

$\leq$ $≤$ Solid Line

$\geq$ $≥$ Solid Line

First, treat the inequality sign as an equals sign and plot the curve.

Graph the line $x = 1$ $x=1$

Since $x < 1$ $x<1$ , we can determine that the shaded part would be the left side because all values on the left side are less than $1$ $1$

Next, graph the line $y = 3 x$ $y=3x$

Use a test point to see which side of the line is to be shaded. We can try the point $($ $($ $2$ $2$ $,$ $,$ $0$ $0$ $)$ $)$

$y$ $y$ $\leq$ $≤$ $3$ $3$ $x$ $x$

$0$ $0$ $\leq$ $≤$ $3 ($ $3($ $2$ $2$ $)$ $)$ Substitute $(2, 0)$ $(2,0)$

$0$ $0$ $\leq$ $≤$ $6$ $6$

The inequality is true, so we will shade the side of the line which includes the point $(2, 0)$ $(2,0)$

Next, graph the curve $y = x^{2}$ $y=x^2$

Use a test point to see which side of the line is to be shaded. We can try the point $($ $($ $0$ $0$ $,$ $,$ $2$ $2$ $)$ $)$

$y$ $y$ $\geq$ $≥$ $\color{#9a00c7}{x}^2$

$2$ $2$ $\geq$ $≥$ $\color{#9a00c7}{0}^2$ Substitute $(0, 2)$ $(0,2)$

$2$ $2$ $\geq$ $≥$ $0$ $0$

The inequality is true, so we will shade the side of the line which includes the point $(0, 2)$ $(0,2)$

Finally, shade the overlapping regions.
Question 3 of 4

3. Question
Graph the intersection of regions

$y \leq x^{2}$ $y≤x^2$

$x \leq 2$ $x≤2$

$y>-1$
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Remember the following notations when graphing inequalities.

Symbol Solid / Dotted

$<$ $<$ Dotted Line

$>$ $>$ Dotted Line

$\leq$ $≤$ Solid Line

$\geq$ $≥$ Solid Line

First, treat the inequality sign as an equals sign and plot the curve.

Graph the line $y = - 1$ $y=-1$

Since $y > - 1$ $y> -1$ , we can determine that the shaded part would be the upper side because all values on the upper side are greater than $- 1$ $-1$

Next, graph the line $x = 2$ $x=2$

Since $x \leq 2$ $x≤2$ , we can determine that the shaded part would be the left side because all values on the left side are less than $2$ $2$

Next, graph the curve $y = x^{2}$ $y=x^2$

Use a test point to see which side of the line is to be shaded. We can try the point $($ $($ $0$ $0$ $,$ $,$ $3$ $3$ $)$ $)$

$y$ $y$ $\leq$ $≤$ $\color{#9a00c7}{x}^2$

$3$ $3$ $\leq$ $≤$ $\color{#9a00c7}{0}^2$ Substitute $(0, 2)$ $(0,2)$

$3$ $3$ $\leq$ $≤$ $0$ $0$

The inequality is false, so we will shade the side of the line which does not include the point $(0, 3)$ $(0,3)$

Finally, shade the overlapping regions.
Question 4 of 4

4. Question
Graph the intersection of regions

$x^{2} + y^{2} \leq 4$ $x^2+y^2≤4$

$y>1$

$x+y>2$
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Remember the following notations when graphing inequalities.

Symbol Solid / Dotted

$<$ $<$ Dotted Line

$>$ $>$ Dotted Line

$\leq$ $≤$ Solid Line

$\geq$ $≥$ Solid Line

First, treat the inequality sign as an equals sign and plot the curve.

Graph the line $x + y = 2$ $x+y=2$

Use a test point to see which side of the line is to be shaded. We can try the origin, $($ $($ $0$ $0$ $,$ $,$ $0$ $0$ $)$ $)$

$\color{#9a00c7}{x}+\color{#00880A}{y}$ $>$ $>$ $2$ $2$

$\color{#9a00c7}{0}+\color{#00880A}{0}$ $>$ $>$ $2$ $2$ Substitute $(0, 0)$ $(0,0)$

$0$ $0$ $>$ $>$ $2$ $2$

The inequality is false, so we will shade the side of the line which does not include the origin

Next, graph the line $y = 1$ $y=1$

Since $y > 1$ $y>1$ , we can determine that the shaded part would be the upper side because all values on the upper side are greater than $1$ $1$

Next, graph the curve $x^{2} + y^{2} = 4$ $x^2+y^2=4$

Use a test point to see which side of the line is to be shaded. We can try the origin, $($ $($ $0$ $0$ $,$ $,$ $0$ $0$ $)$ $)$

$\color{#9a00c7}{x}^2+\color{#00880A}{y}^2$ $\leq$ $≤$ $4$ $4$

$\color{#9a00c7}{0}^2+\color{#00880A}{0}^2$ $\leq$ $≤$ $4$ $4$ Substitute $(0, 0)$ $(0,0)$

$0$ $0$ $\leq$ $≤$ $4$ $4$

The inequality is true, so we will shade the side of the line which includes the origin

Finally, shade the overlapping regions.

Graphing the Intersection of Regions 2

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Quiz summary

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1. Question

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2. Question

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Correct!

3. Question

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4. Question

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Quizzes

Symbol	Solid / Dotted
$<$ $<$	Dotted Line
$>$ $>$	Dotted Line
$\leq$ $≤$	Solid Line
$\geq$ $≥$	Solid Line

$\color{#9a00c7}{x}^2+\color{#00880A}{y}^2$	$\leq$ $≤$	$4$ $4$
$\color{#9a00c7}{0}^2+\color{#00880A}{0}^2$	$\leq$ $≤$	$4$ $4$	Substitute $(0, 0)$ $(0,0)$
$0$ $0$	$\leq$ $≤$	$4$ $4$

$y$ $y$	$\leq$ $≤$	$3$ $3$ $x$ $x$
$0$ $0$	$\leq$ $≤$	$3 ($ $3($ $2$ $2$ $)$ $)$	Substitute $(2, 0)$ $(2,0)$
$0$ $0$	$\leq$ $≤$	$6$ $6$

$y$ $y$	$\geq$ $≥$	$\color{#9a00c7}{x}^2$
$2$ $2$	$\geq$ $≥$	$\color{#9a00c7}{0}^2$	Substitute $(0, 2)$ $(0,2)$
$2$ $2$	$\geq$ $≥$	$0$ $0$

$\color{#9a00c7}{x}+\color{#00880A}{y}$	$>$ $>$	$2$ $2$
$\color{#9a00c7}{0}+\color{#00880A}{0}$	$>$ $>$	$2$ $2$	Substitute $(0, 0)$ $(0,0)$
$0$ $0$	$>$ $>$	$2$ $2$