Algebra 1

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Algebra 1

• 7.6 Solving Systems of Linear Inequalities

Objective
Students will solve systems of linear
inequalities by graphing.
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Before we begin

• In previous lessons we have explored different
  ways to solve systems of linear equations
• In this lesson we will look at linear
  inequalities
• Essentially, you will graph the linear system of
  inequalities on the same coordinate plane, shade
  the solution area for each inequality. The
  portion of the coordinate plane where the shading
  overlaps represents the solution to the system of
  linear inequalities.

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Review

• We have already worked with some of this
  materialas a quick review, you should already
  know that when graphing inequalities
• lt and gt are represented as a dashed line
• and are represented as a solid line
• The shaded portion of the coordinate plane
  represents the solution set to the inequality.
  That is, any point in the shaded area, when
  substituted, will make the inequality true

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Comments

• I cannot stress the importance of being organized
  and laying out your work here
• The same strategies you used to graph equations
  will be used to graph inequalities
• It is not enough to be able to mechanically graph
  the inequalitiesyou are also expected to be able
  to interpret the results
• That is, you must be able to read the graph and
  determine where and what the solution set is
• The key here is to analyze the inequalities first!

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Process

• The process for solving systems of linear
  inequalities is
• Step 1 Write the inequality in a format that is
  easy to graph
• Step 2 Graph and shade the solution set for
  each of the inequalities on the same coordinate
  plane
• Step 3 Identify the area where the shading
  overlaps
• Step 4 Choose a point in the overlapping shaded
  area and substitute it into each of the
  inequalities and determine if you get a true or
  false statement.

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Example 1

• Solve the system of linear inequalities by
  graphing.
• y lt 2 Inequality 1
• x -1 Inequality 2

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Example 1
y lt 2 Inequality 1
Step 1 Write the inequality in a format that is
easy to graph
x -1 Inequality 2
The first step is to analyze the inequalities. I
see that all the inequalities are in a format
that I can easily graphTherefore, I do not need
to do this step.
Something to think aboutIn the back of my mind I
see that inequality 1 2 have only 1
variablefrom working with equations I know that
an equation in 1 variable produces either a
horizontal or vertical linethe same holds true
for inequalitiesI already have a picture of what
the graph will look like in the back of my mind
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Example 1
x -1
Step 2 Graph and shade the solution set for
each of the inequalities on the same coordinate
plane
y lt 2
y lt 2 Inequality 1
x -1 Inequality 2
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Example 1
x -1
Step 3 Identify the area where the shading
overlaps
y lt 2
y lt 2 Inequality 1
x -1 Inequality 2
In this example the square where the 2 solution
sets overlap represents the solution set to the
system of inequalities
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Example 1
Step 4 Choose a point in the overlapping shaded
area and substitute it into each of the
inequalities and determine if you get a true or
false statement
y lt 2 Inequality 1
x -1 Inequality 2
In this example the origin (0, 0) lies within the
solution set. I will use that point to determine
if the solution set is correct by substituting
the values of x and y into the original
inequalities
y lt 2
x -1
0 lt 2
0 -1
True
True
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Comments

• When choosing a point in the overlapping shaded
  area be careful if you choose a point on the
  line
• If the line is dashed ( lt or gt) the points on the
  line are not included in the solution set
• If the line is solid ( or ) the points on the
  line are included in the solution set.

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Example 2

• Sometimes you are given a graph of a system of
  linear inequalities and are asked to write the
  system of inequalities.
• Again, it is expected that you can read the graph
  and determine the inequalities that the graph
  represents
• Lets look at an example

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Example 2

• Write a system of linear inequalities that
  defines the shaded region to the right

Line 1
Line 2
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Example 2
Lets look at Line 1 first
Line 1
In this example I see that Line 1 crosses the
y-axis at 3.
I see that a dashed line is used so I will use
the lt or gt symbol
I also see that the area below the line is
shaded. That means the value is less than.
Therefore, the inequality for line 1 is written
as y lt 3
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Example 2
Now lets look at Line 2
I see that Line 2 crosses the y-axis at 1.
Again, I see that a dashed line is used so I will
use the lt or gt symbol
Line 2
I also see that the area above the line is
shaded. That means the value is greater than.
Therefore, the inequality for line 2 is written
as y gt 1
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Example 2
Line 1
After analyzing the graph we can now determine
the system of inequalities that the graph
represents as y lt 3 y gt 1
Line 2
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Comments

• For some reason students have a hard time with
  reading graphs
• The expectation is if you are given an equation
  or inequality and you know how to graph it using
  slope-intercept formthen you should be able to
  look at a graph, pick out the parts of the
  slope-intercept form and determine the equation
  or inequality of the graph

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Comments

• On the next couple of slides are some practice
  problemsThe answers are on the last slide
• Do the practice and then check your answersIf
  you do not get the same answer you must question
  what you didgo back and problem solve to find
  the error
• If you cannot find the error bring your work to
  me and I will help

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Your Turn

• Write a system of linear inequalities that define
  the shaded regions

2.
3.
1.
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Your Turn

• Graph the system of linear inequalities.
• 2x y gt 2 and 6x 3y lt 12
• 2x 2y lt 6 and x y lt 9
• x 3y 12 and x 6y 12
• x y 6 and x 1 and y 0

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Your Turn

• Graph the system of linear inequalities

8. 3/2x y lt 3 x gt 0
10. -3/2x y 3 4x y lt 2
9. y 0 y 5
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Summary

• A key tool in making learning effective is being
  able to summarize what you learned in a lesson in
  your own words
• In this lesson we talked about systems of linear
  inequalities. Therefore, in your own words
  summarize this lessonbe sure to include key
  concepts that the lesson covered as well as any
  points that are still not clear to you
• I will give you credit for doing this
  lessonplease see the next slide

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Credit

• I will add 25 points as an assignment grade for
  you working on this lesson
• To receive the full 25 points you must do the
  following
• Have your name, date and period as well a lesson
  number as a heading.
• Do each of the your turn problems showing all
  work
• Have a 1 paragraph summary of the lesson in your
  own words
• Please be advised I will not give any credit
  for work submitted
• Without a complete heading
• Without showing work for the your turn problems
• Without a summary in your own words