Graphing the Intersection of Regions 1

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Question 1 of 4

1. Question
Graph the intersection of regions:

$y \leq 2 - x$ $y≤2-x$

$y>x$
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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 $y = x$ $y=x$

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

$y$ $y$ $>$ $>$ $x$ $x$

$0$ $0$ $>$ $>$ $- 2$ $-2$ Substitute $(- 2, 0)$ $(-2,0)$

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

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

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

$y$ $y$ $\leq$ $≤$ $2 -$ $2-$ $x$ $x$

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

$0$ $0$ $\leq$ $≤$ $2$ $2$

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

$2 x + 3 y \leq 6$ $2x+3y≤6$

$x<1$
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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 $2 x + 3 y = 6$ $2x+3y=6$

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

$2$ $2$ $x$ $x$ $+ 3$ $+3$ $y$ $y$ $\leq$ $≤$ $6$ $6$

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

$0$ $0$ $\leq$ $≤$ $6$ $6$

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

Finally, shade the overlapping regions.
Question 3 of 4

3. Question
Graph the intersection of regions

$y \geq x^{2}$ $y≥x^2$

$x+y<2$
- 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 $x + y = 2$ $x+y=2$

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

$x$ $x$ $+$ $+$ $y$ $y$ $<$ $<$ $6$ $6$

$0$ $0$ $+$ $+$ $0$ $0$ $<$ $<$ $6$ $6$ Substitute $(0, 0)$ $(0,0)$

$0$ $0$ $<$ $<$ $6$ $6$

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

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 $(0, 0)$ $(0,0)$

$2$ $2$ $\geq$ $≥$ $0$ $0$

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 4 of 4

4. Question
Graph the intersection of regions

$x>-1$

$y \leq 2 x - 1$ $y≤2x-1$
- 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 right side because all values on the right side are greater than $- 1$ $-1$

Next, graph the line $y = 2 x - 1$ $y=2x-1$

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

$y$ $y$ $\leq$ $≤$ $2$ $2$ $x$ $x$ $- 1$ $-1$

$0$ $0$ $\leq$ $≤$ $2 ($ $2($ $0$ $0$ $) - 1$ $)-1$ Substitute $(0, 0)$ $(0,0)$

$0$ $0$ $\leq$ $≤$ $- 1$ $-1$

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

Finally, shade the overlapping regions.

Graphing the Intersection of Regions 1

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Quiz summary

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1. Question

Hint

Well Done!

2. Question

Hint

Nice Job!

3. Question

Hint

Excellent!

4. Question

Hint

Fantastic!

Quizzes

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

$y$ $y$	$>$ $>$	$x$ $x$
$0$ $0$	$>$ $>$	$- 2$ $-2$	Substitute $(- 2, 0)$ $(-2,0)$

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

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

$x$ $x$ $+$ $+$ $y$ $y$	$<$ $<$	$6$ $6$
$0$ $0$ $+$ $+$ $0$ $0$	$<$ $<$	$6$ $6$	Substitute $(0, 0)$ $(0,0)$
$0$ $0$	$<$ $<$	$6$ $6$

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

$y$ $y$	$\leq$ $≤$	$2$ $2$ $x$ $x$ $- 1$ $-1$
$0$ $0$	$\leq$ $≤$	$2 ($ $2($ $0$ $0$ $) - 1$ $)-1$	Substitute $(0, 0)$ $(0,0)$
$0$ $0$	$\leq$ $≤$	$- 1$ $-1$