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The figure shows two number lines. The number line on the left is labeled x is less than a. The number line shows a parenthesis at a and an arrow that points to the left. The number line on the right is labeled x is less than or equal to a. The number line shows a bracket at a and an arrow that points to the left.

Similarly, for an inequality in two variables, the boundary line is shown with a solid or dashed line to indicate whether or not it the line is included in the solution. This is summarized in [link]

A x + B y < C A x + B y C
A x + B y > C A x + B y C
Boundary line is not included in solution. Boundary line is included in solution.
Boundary line is dashed. Boundary line is solid.

Now, let’s take a look at what we found in [link] . We’ll start by graphing the line y = x + 4 , and then we’ll plot the five points we tested. 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 x plus 4 is plotted as an arrow extending from the bottom left toward the upper right. The following points are plotted and labeled (negative 8, 12), (1, 6), (2, 6), (0, 0), and (negative 5, negative 15).

In [link] we found that some of the points were solutions to the inequality y > x + 4 and some were not.

Which of the points we plotted are solutions to the inequality y > x + 4 ? The points ( 1 , 6 ) and ( −8 , 12 ) are solutions to the inequality y > x + 4 . Notice that they are both on the same side of the boundary line y = x + 4 .

The two points ( 0 , 0 ) and ( −5 , −15 ) are on the other side of the boundary line     y = x + 4 , and they are not solutions to the inequality y > x + 4 . For those two points, y < x + 4 .

What about the point ( 2 , 6 ) ? Because 6 = 2 + 4 , the point is a solution to the equation y = x + 4 . So the point ( 2 , 6 ) is on the boundary line.

Let’s take another point on the left side of the boundary line and test whether or not it is a solution to the inequality y > x + 4 . The point ( 0 , 10 ) clearly looks to be to the left of the boundary line, doesn’t it? Is it a solution to the inequality?

y > x + 4 10 > ? 0 + 4 10 > 4 So , ( 0 , 10 ) is a solution to y > x + 4 .

Any point you choose on the left side of the boundary line is a solution to the inequality y > x + 4 . All points on the left are solutions.

Similarly, all points on the right side of the boundary line, the side with ( 0 , 0 ) and ( −5 , −15 ) , are not solutions to y > x + 4 . 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 x plus 4 is plotted as an arrow extending from the bottom left toward the upper right. The following points are plotted and labeled (negative 8, 12), (1, 6), (2, 6), (0, 0), and (negative 5, negative 15). To the upper left of the line is the inequality y is greater than x plus 4. To the right of the line is the inequality y is less than x plus 4.

The graph of the inequality y > x + 4 is shown in [link] below. The line y = x + 4 divides the plane into two regions. The shaded side shows the solutions to the inequality y > x + 4 .

The points on the boundary line, those where y = x + 4 , are not solutions to the inequality y > x + 4 , so the line itself is not part of the solution. We show that by making the line dashed, not solid.

The graph shows the x y-coordinate plane. The x- and y-axes each run from negative 10 to 10. The line y equals x plus 4 is plotted as a dashed arrow extending from the bottom left toward the upper right. The coordinate plane to the upper left of the line is shaded.
The graph of the inequality y > x + 4 .

The boundary line shown is y = 2 x 1 . 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 y equals 2 x minus 1 is plotted as a solid arrow extending from the bottom left toward the upper right. The coordinate plane to the left of the line is shaded

Solution

The line y = 2 x 1 is the boundary line. On one side of the line are the points with y > 2 x 1 and on the other side of the line are the points with y < 2 x 1 .

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:

y > 2 x 1 or y < 2 x 1 ? y > 2 x 1 y < 2 x 1 0 > ? 2 · 0 1 0 < ? 2 · 0 1 0 > −1 True 0 < −1 False

Since, y > 2 x 1 is true, the side of the line with ( 0 , 0 ) , is the solution. The shaded region shows the solution of the inequality y > 2 x 1 .

Since the boundary line is graphed with a solid line, the inequality includes the equal sign.

The graph shows the inequality y 2 x 1 .

We could use any point as a test point, provided it is not on the line. Why did we choose ( 0 , 0 ) ? Because it’s the easiest to evaluate. You may want to pick a point on the other side of the boundary line and check that y < 2 x 1 .

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Write the inequality shown by the graph with the boundary line y = −2 x + 3 .

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 plus 3 is plotted as a solid arrow extending from the top left toward the bottom right. The coordinate plane to the right of the line is shaded.

y −2 x + 3

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Write the inequality shown by the graph with the boundary line y = 1 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 one half x minus 4 is plotted as a solid arrow extending from the bottom left toward the top right. The coordinate plane to the bottom right of the line is shaded.

y < 1 2 x 4

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Questions & Answers

how did you get 1640
Noor Reply
If auger is pair are the roots of equation x2+5x-3=0
Peter Reply
Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.
MATTHEW Reply
420
Sharon
from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph
George
Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28
Salma
Devon is 32 32​​ years older than his son, Milan. The sum of both their ages is 54 54​. Using the variables d d​ and m m​ to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.
Aaron Reply
find product (-6m+6) ( 3m²+4m-3)
SIMRAN Reply
-42m²+60m-18
Salma
what is the solution
bill
how did you arrive at this answer?
bill
-24m+3+3mÁ^2
Susan
i really want to learn
Amira
I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please
Amira
Hw did u arrive to this answer.
Aphelele
hi
Bajemah
-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18
Salma
complete the table of valuesfor each given equatio then graph. 1.x+2y=3
Jovelyn Reply
x=3-2y
Salma
y=x+3/2
Salma
Hi
Enock
given that (7x-5):(2+4x)=8:7find the value of x
Nandala
3x-12y=18
Kelvin
please why isn't that the 0is in ten thousand place
Grace Reply
please why is it that the 0is in the place of ten thousand
Grace
Send the example to me here and let me see
Stephen
A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg
Marry Reply
how far
Abubakar
cool u
Enock
state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°
Abegail Reply
hello
BenJay
hi
Method
I am eliacin, I need your help in maths
Rood
how can I help
Sir
hmm can we speak here?
Amoon
however, may I ask you some questions about Algarba?
Amoon
hi
Enock
what the last part of the problem mean?
Roger
The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.
cameron Reply
Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.
mahnoor Reply
I'm guessing, but it's somewhere around $4335.00 I think
Lewis
12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00
Munster
difference between rational and irrational numbers
Arundhati Reply
When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?
Jakoiya Reply
how to reduced echelon form
Solomon Reply
Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?
Zack Reply
d=r×t the equation would be 8/r+24/r+4=3 worked out
Sheirtina
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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