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Graph the linear inequality y < 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 5 is plotted as a dashed arrow horizontally across the plane. The region above the line is shaded.

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Graph the linear inequality y −1 .

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 1 is plotted as a dashed arrow horizontally across the plane. The region below the line is shaded.

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Key concepts

  • To 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.

Practice makes perfect

Verify Solutions to an Inequality in Two Variables

In the following exercises, determine whether each ordered pair is a solution to the given inequality.

Determine whether each ordered pair is a solution to the inequality y > x 1 :

( 0 , 1 )
( −4 , −1 )
( 4 , 2 )
( 3 , 0 )
( −2 , −3 )

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Determine whether each ordered pair is a solution to the inequality y > x 3 :

( 0 , 0 )
( 2 , 1 )
( −1 , −5 )
( −6 , −3 )
( 1 , 0 )

yes  no  no  yes  no

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Determine whether each ordered pair is a solution to the inequality y < x + 2 :

( 0 , 3 )
( −3 , −2 )
( −2 , 0 )
( 0 , 0 )
( −1 , 4 )

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Determine whether each ordered pair is a solution to the inequality y < x + 5 :

( −3 , 0 )
( 1 , 6 )
( −6 , −2 )
( 0 , 1 )
( 5 , −4 )

yes  no  no  yes  yes

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Determine whether each ordered pair is a solution to the inequality x + y > 4 :

( 5 , 1 )
( −2 , 6 )
( 3 , 2 )
( 10 , −5 )
( 0 , 0 )

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Determine whether each ordered pair is a solution to the inequality x + y > 2 :

( 1 , 1 )
( 4 , −3 )
( 0 , 0 )
( −8 , 12 )
( 3 , 0 )

no  no  no  yes  yes

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Recognize the Relation Between the Solutions of an Inequality and its Graph

In the following exercises, write the inequality shown by the shaded region.

Write the inequality shown by the graph with the boundary line y = 3 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 3x minus 4 is plotted as a dashed line extending from the bottom left toward the top right. The region to the right of the line is shaded.
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Write the inequality shown by the graph with the boundary line y = 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 2x minus 4 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.

y < 2 x 4

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Write the inequality shown by the graph with the boundary line y = 1 2 x + 1 .

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 one-half x plus 1 is plotted as a solid line extending from the bottom left toward the top right. The region below the line is shaded.
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Write the inequality shown by the graph with the boundary line y = 1 3 x 2 .

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 one-third x minus 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.

y 1 3 x 2

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Write the inequality shown by the shaded region in the graph with the boundary line x + y = 5 .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line x plus y equals 5 is plotted as a solid line extending from the top left toward the bottom right. The region above the line is shaded.
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Write the inequality shown by the shaded region in the graph with the boundary line 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 x plus y equals 3 is plotted as a solid line extending from the top left toward the bottom right. The region above the line is shaded.

x + y 3

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Write the inequality shown by the shaded region in the graph with the boundary line 2 x + y = −4 .

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 y equals negative 4 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.
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Write the inequality shown by the shaded region in the graph with the boundary line x + 2 y = −2 .

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

x + 2 y −2

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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 dashed line extending from the bottom left toward the top right. The region to the left of the line is shaded.
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Write the inequality shown by the shaded region in the graph with the boundary line 2 x y = 4 .

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 4 is plotted as a dashed line extending from the bottom left toward the top right. The region to the left of the line is shaded.

2 x y < 4

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Write the inequality shown by the shaded region in the graph with the boundary line 2 x 5 y = 10 .

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 5 y equals 10 is plotted as a dashed line extending from the bottom left toward the top right. The region below the line is shaded.
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Write the inequality shown by the shaded region in the graph with the boundary line 4 x 3 y = 12 .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 4 x minus 3 y equals 12 is plotted as a dashed line extending from the bottom left toward the top right. The region below the line is shaded.

4 x 3 y > 12

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Graph Linear Inequalities

In the following exercises, graph each linear inequality.

Graph the linear inequality y > 2 3 x 1 .

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Graph the linear inequality y < 3 5 x + 2 .

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

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Graph the linear inequality y 1 2 x + 4 .

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Graph the linear inequality y 1 3 x 2 .

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 one-third x minus 2 is plotted as a solid line extending from the top left toward the bottom right. The region below the line is shaded.

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Graph the linear inequality x y 3 .

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Graph the linear inequality x y −2 .

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

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Graph the linear inequality 4 x + y > −4 .

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Graph the linear inequality x + 5 y < −5 .

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

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Graph the linear inequality 3 x + 2 y −6 .

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Graph the linear inequality 4 x + 2 y −8 .

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line 4 x plus 2 y equals negative 8 is plotted as a solid line extending from the top left toward the bottom right. The region to the right of the line is shaded.

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Questions & Answers

Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hour longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Markeice Reply
8mph
michele
16mph
Robert
3.8 mph
Ped
16 goes into 80 5times while 20 goes into 80 4times and is 4mph faster
Robert
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
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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