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The boundary line shown is 2 x + 3 y = 6 . Write the inequality shown by the graph.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x plus 3 y equals 6 is plotted as a dashed arrow extending from the top left toward the bottom right. The coordinate plane to the bottom of the line is shaded.

The line 2 x + 3 y = 6 is the boundary line. On one side of the line are the points with 2 x + 3 y > 6 and on the other side of the line are the points with 2 x + 3 y < 6 .

Let’s test the point ( 0 , 0 ) and see which inequality describes its side of the boundary line.

At ( 0 , 0 ) , which inequality is true:

2 x + 3 y > 6 or 2 x + 3 y < 6 ? 2 x + 3 y > 6 2 x + 3 y < 6 2 ( 0 ) + 3 ( 0 ) > ? 6 2 ( 0 ) + 3 ( 0 ) < ? 6 0 > 6 False 0 < 6 True

So the side with ( 0 , 0 ) is the side where 2 x + 3 y < 6 .

(You may want to pick a point on the other side of the boundary line and check that 2 x + 3 y > 6 .)

Since the boundary line is graphed as a dashed line, the inequality does not include an equal sign.

The graph shows the solution to the inequality 2 x + 3 y < 6 .

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Write the inequality shown by the shaded region in the graph with the boundary line x 4 y = 8 .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus 4 y equals 8 is plotted as a solid arrow extending from the bottom left toward the top right. The coordinate plane to the top of the line is shaded.

x 4 y 8

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Write the inequality shown by the shaded region in the graph with the boundary line 3 x y = 6 .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 3 x minus y equals 6 is plotted as a solid arrow extending from the bottom left toward the top right. The coordinate plane to the right of the line is shaded.

3 x y 6

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Graph linear inequalities

Now, we’re ready to put all this together to graph linear inequalities.

How to graph linear inequalities

Graph the linear inequality     y 3 4 x 2 .

Solution

This figure is a table that has three columns and three rows. The first column is a header column, and it contains the names and numbers of each step. The second column contains further written instructions. The third column contains math. On the top row of the table, the first cell on the left reads: “Step 1. Identify and graph the boundary line. If the inequality is less than or equal to or greater than or equal to, the boundary line is solid. If the inequality is less than or greater than, the boundary line is dashed. The text in the second cell reads: “Replace the inequality sign with an equal sign to find the boundary line. Graph the boundary line y equals three-fourths x minus 2. The inequality sign is greater than or equal to, so we draw a solid line. The third cell contains the graph of the line three-fourths x minus 2 on a coordinate plane. In the second row of the table, the first cell says: “Step 2. Test a point that is not on the boundary line. Is it a solution of the inequality? In the second cell, the instructions say: “We’ll test (0, 0). Is it a solution of the inequality?” The third cell asks: At (0, 0), is y greater than or equal to three-fourths x minus 2? Below that is the inequality 0 is greater than or equal to three-fourths 0 minus 2, with a question mark above the inequality symbol. Below that is the inequality 0 is greater than or equal to negative 2. Below that is: “So (0, 0) is a solution. In the third row of the table, the first cell says: “Step 3. Shade in one side of the boundary line. If the test point is a solution, shade in the side that includes the point. If the test point is not a solution, shade in the opposite side. In the second cell, the instructions say: The test point (0, 0) is a solution to y is greater than or equal to three-fourths x minus 2. So we shade in that side.” In the third cell is the graph of the line three-fourths x minus 2 on a coordinate plane with the region above the line shaded.
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Graph the linear inequality y 5 2 x 4 .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals five-halves x minus 4 is plotted as a solid arrow extending from the bottom left toward the top right. The region above the line is shaded.

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Graph the linear inequality y < 2 3 x 5 .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals two-thirds x minus 5 is plotted as a dashed arrow extending from the bottom left toward the top right. The region below the line is shaded.

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The steps we take to graph a linear inequality are summarized here.

Graph a linear inequality.

  1. Identify and graph the boundary line.
    • If the inequality is or , the boundary line is solid.
    • If the inequality is<or>, the boundary line is dashed.
  2. Test a point that is not on the boundary line. Is it a solution of the inequality?
  3. Shade in one side of the boundary line.
    • If the test point is a solution, shade in the side that includes the point.
    • If the test point is not a solution, shade in the opposite side.

Graph the linear inequality x 2 y < 5 .

Solution

First we graph the boundary line x 2 y = 5 . The inequality is < so we draw a dashed line.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus 2 y equals 5 is plotted as a dashed arrow extending from the bottom left toward the top right.

Then we test a point. We’ll use ( 0 , 0 ) again because it is easy to evaluate and it is not on the boundary line.

Is ( 0 , 0 ) a solution of x 2 y < 5 ?
The figure shows the inequality 0 minus 2 times 0 in parentheses is less than 5, with a question mark above the inequality symbol. The next line shows 0 minus 0 is less than 5, with a question mark above the inequality symbol. The third line shows 0 is less than 5.

The point ( 0 , 0 ) is a solution of x 2 y < 5 , so we shade in that side of the boundary line.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x minus 2 y equals 5 is plotted as a dashed arrow extending from the bottom left toward the top right. The point (0, 0) is plotted, but not labeled. The region above the line is shaded.

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Graph the linear inequality 2 x 3 y 6 .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x minus 3 y equals 6 is plotted as a solid arrow extending from the bottom left toward the top right. The region above the line is shaded.

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Graph the linear inequality 2 x y > 3 .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 2 x minus y equals 3 is plotted as a dashed arrow extending from the bottom left toward the top right. The region below the line is shaded.

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What if the boundary line goes through the origin? Then we won’t be able to use ( 0 , 0 ) as a test point. No problem—we’ll just choose some other point that is not on the boundary line.

Graph the linear inequality y −4 x .

Solution

First we graph the boundary line y = −4 x . It is in slope–intercept form, with m = −4 and b = 0 . The inequality is so we draw a solid line.
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line s y equals negative 4 x is plotted as a solid arrow extending from the top left toward the bottom right.

Now, we need a test point. We can see that the point ( 1 , 0 ) is not on the boundary line.

Is ( 1 , 0 ) a solution of y −4 x ?
The figure shows 0 is less than or equal to negative 4 times 1 in parentheses, with a question mark above the inequality symbol. The next line shows 0 is not less than or equal to negative 4.

The point ( 1 , 0 ) is not a solution to y −4 x , so we shade in the opposite side of the boundary line. See [link] .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 4 x is plotted as a solid arrow extending from the top left toward the bottom right. The point (1, 0) is plotted, but not labeled. The region to the left of the line is shaded.
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Graph the linear inequality y > −3 x .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 3 x is plotted as a dashed arrow extending from the top left toward the bottom right. The region to the right of the line is shaded.

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Graph the linear inequality y −2 x .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals negative 2 x is plotted as a solid arrow extending from the top left toward the bottom right. The region to the right of the line is shaded.

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Some linear inequalities have only one variable. They may have an x but no y , or a y but no x . In these cases, the boundary line will be either a vertical or a horizontal line. Do you remember?

x = a vertical line y = b horizontal line

Graph the linear inequality y > 3 .

Solution

First we graph the boundary line y = 3 . It is a horizontal line. The inequality is>so we draw a dashed line.

We test the point ( 0 , 0 ) .

y > 3 0 > 3

( 0 , 0 ) is not a solution to y > 3 .

So we shade the side that does not include (0, 0).
The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals 3 is plotted as a dashed arrow horizontally across the plane. The region above the line is shaded.

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Questions & Answers

what is the answer for this 3×9+28÷4-8
What Reply
315
lashonna
how do you do xsquard+7x+10=0
What
(x + 2)(x + 5), then set each factor to zero and solve for x. so, x = -2 and x = -5.
bruce
I skipped it
What
In 10 years, the population of Detroit fell from 950,000 to about 712,500. Find the percent decrease.
Jenise Reply
how do i set this up
Jenise
25%
Melissa
25 percent
Muzamil
950,000 - 712,500 = 237,500. 237,500 / 950,000 = .25 = 25%
Melissa
I've tried several times it won't let me post the breakdown of how you get 25%.
Melissa
Subtract one from the other to get the difference. Then take that difference and divided by 950000 and you will get .25 aka 25%
Melissa
Finally 👍
Melissa
one way is to set as ratio: 100%/950000 = x% / 712500, which yields that 712500 is 75% of the initial 950000. therefore, the decrease is 25%.
bruce
twenty five percent...
Jeorge
thanks melissa
Jeorge
950000-713500 *100 and then divide by 950000 = 25
Muzamil
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
jamie Reply
6t+3
Melissa
6t +3
Bollywood
Tricia got a 6% raise on her weekly salary. The raise was $30 per week. What was her original salary?
Iris Reply
let us suppose her original salary is 'm'. so, according to the given condition, m*(6/100)=30 m= (30*100)/6 m= 500 hence, her original salary is $500.
Simply
28.50
Toi
thanks
Jeorge
How many pounds of nuts selling for $6 per pound and raisins selling for $3 per pound should Kurt combine to obtain 120 pounds of trail mix that cost him $5 per pound?
Valeria Reply
Amber wants to put tiles on the backsplash of her kitchen counters. She will need 36 square feet of tiles. She will use basic tiles that cost $8 per square foot and decorator tiles that code $20 per square foot. How many square feet of each tile should she use so that the overal cost of he backsplash will be $10 per square foot?
Nia Reply
I need help with maths can someone help me plz.. is there a wats app group?
Cindy Reply
WY need
Fernando
How did you get $750?
Laura Reply
if y= 2x+sinx what is dy÷dx
formon25 Reply
does it teach you how to do algebra if you don't know how
Kate Reply
Liam borrowed a total of $35,000 to pay for college. He pays his parents 3% interest on the $8,000 he borrowed from them and pays the bank 6.8% on the rest. What average interest rate does he pay on the total $35,000? (Round your answer to the nearest tenth of a percent.)
Lara Reply
exact definition of length by bilbao
francemichael Reply
the definition of length
francemichael Reply
literal meaning of length
francemichael
exact meaning of length
francemichael
exact meaning of length
francemichael
how many typos can we find...?
Chris Reply
5
Joseph
In the LCM Prime Factors exercises, the LCM of 28 and 40 is 280. Not 420!
Chris Reply
Practice Key Terms 3

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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