Graph Linear Inequalities 1

Topics

Algebra 1

Inequalities

Graph Linear Inequalities

Graph Linear Inequalities 1

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Question 1 of 8

1. Question
Graph $x > 3$ $x > 3$
- 1.
- 2.
- 3.
- 4.
Hint

Help Video

Correct

Well Done!

Incorrect

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 $x = 3$ $x=3$

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

$x$ $x$ $>$ $>$ $3$ $3$

$0$ $0$ $>$ $>$ $3$ $3$ Plug in $0$ $0$ as $x$ $x$

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

Now we can graph the inequality.

$>$ $>$ means we must use a dashed line when graphing $x > 3$ $x > 3$
Question 2 of 8

2. Question
Graph $y \leq 2$ $y ≤ 2$
- 1.
- 2.
- 3.
- 4.
Hint

Help Video

Correct

Nice Job!

Incorrect

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 $y = 2$ $y=2$

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

$y$ $y$ $\leq$ $≤$ $2$ $2$

$0$ $0$ $\leq$ $≤$ $2$ $2$ Plug in $0$ $0$ as $y$ $y$

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

Now we can graph the inequality

$\leq$ $≤$ means we must use a solid line when graphing $y \leq 2$ $y ≤ 2$
Question 3 of 8

3. Question
Graph $x > 4$ $x > 4$
- 1.
- 2.
- 3.
- 4.
Hint

Help Video

Correct

Excellent!

Incorrect

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 $x = 4$ $x=4$

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

$x$ $x$ $>$ $>$ $4$ $4$

$0$ $0$ $>$ $>$ $4$ $4$ Plug in $0$ $0$ as $x$ $x$

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

Now we can graph the inequality.

$>$ $>$ means we must use a dashed line when graphing $x > 4$ $x > 4$
Question 4 of 8

4. Question
Graph $y > 2 x - 3$ $y > 2x-3$
- 1.
- 2.
- 3.
- 4.
Hint

Help Video

Correct

Fantastic!

Incorrect

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 $y =$ $y=$ $2$ $2$ $x$ $x$ $- 3$ $-3$

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

$y$ $y$ $>$ $>$ $2$ $2$ $x$ $x$ $- 3$ $-3$

$0$ $0$ $>$ $>$ $2$ $2$ $(0)$ $(0)$ $- 3$ $-3$ Plug in $0$ $0$ as $x$ $x$ and $y$ $y$

$0$ $0$ $>$ $>$ $- 3$ $-3$ Simplify

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

Now we can graph the inequality.

$>$ $>$ means we must use a dashed line when graphing $y > 2 x - 3$ $y > 2x-3$
Question 5 of 8

5. Question
Graph $y \leq - 3 x + 2$ $y ≤ -3x+2$
- 1.
- 2.
- 3.
- 4.
Hint

Help Video

Correct

Keep Going!

Incorrect

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 $y =$ $y=$ $- 3$ $-3$ $x +$ $x+$ $2$ $2$

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

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

$0$ $0$ $\leq$ $≤$ $- 3$ $-3$ $(0)$ $(0)$ $+ 2$ $+2$ Plug in $0$ $0$ as $x$ $x$ and $y$ $y$

$0$ $0$ $\leq$ $≤$ $2$ $2$ Simplify

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

Now we can graph the inequality

$\leq$ $≤$ means we must use a solid line when graphing $y \leq - 3 x + 2$ $y ≤ -3x+2$

Question 6 of 8

Graph

2 x + 3 y > 6

$2x+3y > 6$

Incorrect

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

2 x + 3 y = 6

$2x+3y=6$

Write the equation in point-gradient form.

$2 x + 3 y$ $2x+3y$	$=$ $=$	$6$ $6$
$2 x + 3 y$ $2x+3y$ $- 2 x$ $-2x$	$=$ $=$	$6$ $6$ $- 2 x$ $-2x$	Subtract $2 x$ $2x$ from both sides
$3 y$ $3y$	$=$ $=$	$- 2 x + 6$ $-2x+6$	Simplify
$y$ $y$	$=$ $=$	$- \frac{2}{3} x + 2$ $-2/3x+2$	Divide both sides by $3$ $3$

You may now graph the line

2 x + 3 y = 6

$2x+3y=6$ with its gradient-intercept form.

The slope of the line is $- \frac{2}{3}$ $-2/3$ and the y-intercept is $2$ $2$ .

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

(

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

)

$)$ .

$2$ $2$ $x$ $x$ $+ 3$ $+3$ $y$ $y$	$>$ $>$	$6$ $6$
$2$ $2$ $(0)$ $(0)$ $+ 3$ $+3$ $(0)$ $(0)$	$>$ $>$	$6$ $6$	Plug in $0$ $0$ as $x$ $x$ and $y$ $y$
$0$ $0$	$>$ $>$	$6$ $6$	Simplify

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

Now we can graph the inequality.

>

$>$ means we must use a dashed line when graphing

2 x + 3 y > 6

$2x+3y > 6$

Question 7 of 8

Graph

x + 3 y - 3

$x+3y-3$

<

$<$

0

$0$ and

x - y + 2 \leq 0

$x-y+2 ≤ 0$

Incorrect

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 + 3 y - 3 = 0

$x+3y-3=0$

Write the equation in point-gradient form.

$x + 3 y - 3$ $x+3y-3$	$=$ $=$	$0$ $0$
$x + 3 y - 3$ $x+3y-3$ $- x + 3$ $-x+3$	$=$ $=$	$0$ $0$ $- x + 3$ $-x+3$	Add $- x + 3$ $-x+3$ to both sides
$3 y$ $3y$	$=$ $=$	$- x + 3$ $-x+3$	Simplify
$y$ $y$	$=$ $=$	$- \frac{1}{3} x + 1$ $-1/3x+1$	Divide both sides by $3$ $3$

You may now graph the line

x + 3 y - 3

$x+3y-3$

<

$<$

0

$0$ with its gradient-intercept form.

The slope of the line is $- \frac{1}{3}$ $-1/3$ and the y-intercept is $1$ $1$ .

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

(

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

)

$)$ .

$x$ $x$ $+ 3$ $+3$ $y$ $y$ $- 3$ $-3$	$<$ $<$	$0$ $0$
$0$ $0$ $+ 3$ $+3$ $(0)$ $(0)$ $- 3$ $-3$	$<$ $<$	$0$ $0$	Plug $0$ $0$ as $x$ $x$ and $y$ $y$
$- 3$ $-3$	$<$ $<$	$0$ $0$	Simplify

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

Graph the line

x - y + 2 = 0

$x-y+2=0$

Write the equation in point-gradient form.

$x - y + 2$ $x-y+2$	$=$ $=$	$0$ $0$
$x - y + 2$ $x-y+2$ $- x - 2$ $-x-2$	$=$ $=$	$0$ $0$ $- x - 2$ $-x-2$	Add $- x - 2$ $-x-2$ from both sides
$- y$ $-y$	$=$ $=$	$- x - 2$ $-x-2$	Simplify
$y$ $y$	$=$ $=$	$x + 2$ $x+2$	Divide both sides by $- 1$ $-1$

You may now graph the line

x - y + 2 \leq 0

$x-y+2 ≤ 0$ with its gradient-intercept form.

The slope of the line is $1$ $1$ and the y-intercept is $2$ $2$ .

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

(

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

)

$)$ .

$x$ $x$ $-$ $-$ $y$ $y$ $+ 2$ $+2$	$\leq$ $≤$	$0$ $0$
$0$ $0$ $-$ $-$ $0$ $0$ $+ 2$ $+2$	$\leq$ $≤$	$0$ $0$	Plug in $0$ $0$ as $x$ $x$ and $y$ $y$
$2$ $2$	$\leq$ $≤$	$0$ $0$	Simplify

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

Now we can graph the two inequalities.

<

$<$ means we must use a dashed line when graphing

x + 3 y - 3

$x+3y-3$

<

$<$

0

$0$

\leq

$≤$ means we must use a solid line when graphing

x - y + 2 \leq 0

$x-y+2 ≤ 0$

Question 8 of 8

8. Question
Shade the region bounded by inequalities $y \geq \frac{1}{2} x$ $y≥1/2x$ , $y \leq 2$ $y≤2$ and $y$ $y$ $<$ $<$ $2 x + 4$ $2x+4$
- 1.
- 2.
- 3.
- 4.
Hint

Help Video

Correct

Exceptional!

Incorrect

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 $y = \frac{1}{2} x$ $y=1/2x$

The slope of the line is $\frac{1}{2}$ $1/2$ and the y-intercept is $0$ $0$ .

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

$y$ $y$ $\geq$ $≥$ $\frac{1}{2}$ $1/2$ $x$ $x$

$0$ $0$ $\geq$ $≥$ $\frac{1}{2}$ $1/2$ $(0)$ $(0)$ Plug $0$ $0$ as $x$ $x$ and $y$ $y$

$0$ $0$ $\geq$ $≥$ $0$ $0$ Simplify

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

$\geq$ $≥$    means we must use a solid line when graphing $y \geq \frac{1}{2} x$ $y≥1/2x$

Graph the line $y = 2$ $y=2$ .

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

$y$ $y$ $\leq$ $≤$ $2$ $2$

$0$ $0$ $\leq$ $≤$ $2$ $2$ Plug $0$ $0$ as $y$ $y$

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

$\leq$ $≤$    means we must use a solid line when graphing $y \leq 2$ $y≤2$

Graph the line $y = 2 x + 4$ $y=2x+4$

The slope of the line is $2$ $2$ and the y-intercept is $4$ $4$ .

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

$y$ $y$ $<$ $<$ $2$ $2$ $x$ $x$ $+ 4$ $+4$

$0$ $0$ $<$ $<$ $2$ $2$ $(0)$ $(0)$ $+ 4$ $+4$ Plug $0$ $0$ as $x$ $x$ and $y$ $y$

$0$ $0$ $<$ $<$ $4$ $4$ Simplify

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

$<$ $<$    means we must use a dashed line when graphing $y$ $y$ $<$ $<$ $2 x + 4$ $2x+4$

Now we can graph the three inequalities.

Graph Linear Inequalities 1

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Quiz summary

Information

1. Question

Hint

Well Done!

2. Question

Hint

Nice Job!

3. Question

Hint

Excellent!

4. Question

Hint

Fantastic!

5. Question

Hint

Keep Going!

6. Question

Hint

Correct!

7. Question

Hint

Great Work!

8. Question

Hint

Exceptional!

Quizzes

$x$ $x$	$>$ $>$	$3$ $3$
$0$ $0$	$>$ $>$	$3$ $3$	Plug in $0$ $0$ as $x$ $x$

$x$ $x$	$>$ $>$	$4$ $4$
$0$ $0$	$>$ $>$	$4$ $4$	Plug in $0$ $0$ as $x$ $x$

$y$ $y$	$>$ $>$	$2$ $2$ $x$ $x$ $- 3$ $-3$
$0$ $0$	$>$ $>$	$2$ $2$ $(0)$ $(0)$ $- 3$ $-3$	Plug in $0$ $0$ as $x$ $x$ and $y$ $y$
$0$ $0$	$>$ $>$	$- 3$ $-3$	Simplify

$y$ $y$	$\leq$ $≤$	$- 3$ $-3$ $x$ $x$ $+ 2$ $+2$
$0$ $0$	$\leq$ $≤$	$- 3$ $-3$ $(0)$ $(0)$ $+ 2$ $+2$	Plug in $0$ $0$ as $x$ $x$ and $y$ $y$
$0$ $0$	$\leq$ $≤$	$2$ $2$	Simplify

$y$ $y$	$\geq$ $≥$	$\frac{1}{2}$ $1/2$ $x$ $x$

$0$ $0$	$\geq$ $≥$	$\frac{1}{2}$ $1/2$ $(0)$ $(0)$	Plug $0$ $0$ as $x$ $x$ and $y$ $y$

$0$ $0$	$\geq$ $≥$	$0$ $0$	Simplify

$y$ $y$	$<$ $<$	$2$ $2$ $x$ $x$ $+ 4$ $+4$
$0$ $0$	$<$ $<$	$2$ $2$ $(0)$ $(0)$ $+ 4$ $+4$	Plug $0$ $0$ as $x$ $x$ and $y$ $y$
$0$ $0$	$<$ $<$	$4$ $4$	Simplify