graph linear inequalities in two variables
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Graph linear inequalities in two variables
Section 6.7#44 There is nothing
strange in the circle being the origin of any and
every marvel. Aristotle
Concept Up until this point we’ve discussed inequalities that
involve only one dimension or one variable Today we’re going to take our understanding of
inequalities and apply it to two dimensions (variables) First we will do a short review of lines and linear terms
Slope Slope is
A. An index of the angle of a lineB. A ratio of how much a line increases versus how much to moves right of leftC. A ratio of run to riseD. An index of movement in the x direction
Slope What is the slope of the line that goes through the
points (1,2) & (5,4)
4 3 22;(7, 2)x y
12
12
.
. 2
.
.1
ABCD
Slope What is the slope of the line that goes through the
points (-4,-2) & (7,-8)
4 3 22;(7, 2)x y
1011
512
103
611
.
.
.
.
ABCD
Slope What is the slope of the line that goes through the
points & 1 25 3,
53
157
. 3
.
.
.3
ABCD
6 75 3,
Slope What is the slope of the line that goes through the
points (5,2) & (5,4)
35
15
.
.0
.
.3
ABCD
Slope What is the equation of the line that goes through
the points (1,3) & (3,7)
5 12 2
5 192 2
. 2 1
. 2 13
.
.
A y xB y xC y xD y x
Slope What is the equation of the line that goes through
the points (-3,4) & (-5,-12)
. 1
. 7
. 8 28
. 8 20
A y xB y xC y xD y x
Slope What is the equation of the vertical line that goes
through the point (3,-5)
. 5
. 5
. 3
. 3
A yB xC yD x
Slope The equation of a line is y=3x-9. The slope of the
line is increased by 2. What happens to the line?
A. The line has the same y-intercept, but now slopes downwardB. The line has the same y-intercept, but is now steeperC. The line has a different y-intercept, but now slopes downwardD. The line has a different y-intercept, but is now steeper
Slope Assuming that the line starts at x=0, which line
will reach y=50 first?
. 4 5
. 4 30
. 8
. 8 5
A y xB y xC y xD y x
The big idea When we look at a line, we’re seeing the collection of
points that are solutions to a linear equality When looking at a linear inequality, instead of looking
at a set of points, we are seeing a defined space that indicates the infinite collection of points that satisfy the criteria
For example22 xy
Y
X22 xy
This means that any point that falls in the shaded
area is a viable solution to the inequality
Testing a point We can see this by testing out a point in the shaded
area For example
!103
21232)6(23
Works
Y
X
22 xy
(-6,3)
It’s imperative that we remember that
the solution to these inequalities is an area as opposed
to a line
Process out of examples Our process for creating these graphs is not difficult,
but rather just an extension of our previous knowledge of graphing
Y
X
Graph the line via linear graphing methodsDraw a
dashed line for >,<
otherwise a solid lineShade the appropriat
e areaAbove for greater
thanBelow for less than
Example Let’s do an example
Y
X
43 xy
Example How would we graph this one?
Y
X
6y
Example We would operate horizontal and vertical inequalities
the same as any other inequalityY
X
4x
ExamplesY
X
2 47
y x
Example And another one
Y
X
321
12241242
xy
xyyx
Example And another one
Y
X
9 3 12x y
Practical ExampleA party shop makes giftbags for birthday parties. They charge $4 per glowstick and $10 per T-shirt. Let x represent the number of glowsticks and y the number of T-shirts. The goal is to earn at least $500 from the sale of the bags• Write an inequality that describes the goal in terms of x & y• Graph the inequality• Give three possible combinations of pairs that will allow the shop to meet it’s goal
Y
X
Most Important Points What’s the most important thing that we can learn
from today? The solution to an inequality in two-dimensions is an area, as
opposed to a line We can graph the solutions to an equation by following our
normal processes for graphing lines and then shading the appropriate area
Homework6.7 you will have two days
1, 2-32, 47-50, 53-57
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