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§ 2.4 Graphing Inequalities
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Why We Need This
Our applications will have associated limiting values - and either wewill have to be at least as big as the value or no larger than the value.
These situations are represented by inequalities, and when we take allof those in a problem together, we are looking for the commonintersection. This will be called the feasible set.
The difference between how you would have learned these before andhow we will do this here is that we will shade where the inequality isfalse. You have been taught to always shade where the inequality istrue.
![Page 3: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/3.jpg)
Why We Need This
Our applications will have associated limiting values - and either wewill have to be at least as big as the value or no larger than the value.
These situations are represented by inequalities, and when we take allof those in a problem together, we are looking for the commonintersection. This will be called the feasible set.
The difference between how you would have learned these before andhow we will do this here is that we will shade where the inequality isfalse. You have been taught to always shade where the inequality istrue.
![Page 4: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/4.jpg)
Why We Need This
Our applications will have associated limiting values - and either wewill have to be at least as big as the value or no larger than the value.
These situations are represented by inequalities, and when we take allof those in a problem together, we are looking for the commonintersection. This will be called the feasible set.
The difference between how you would have learned these before andhow we will do this here is that we will shade where the inequality isfalse. You have been taught to always shade where the inequality istrue.
![Page 5: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/5.jpg)
One Inequality
ExampleGraph y ≥ 2x + 1.
First, we plot the linear function associated with the inequality.
-3 -2 -1 321-1
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1
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3
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One Inequality
ExampleGraph y ≥ 2x + 1.
First, we plot the linear function associated with the inequality.
-3 -2 -1 321-1
-2
-3
1
2
3
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One Inequality
Then, we decide which side of the line satisfies the inequality. Theeasiest way I know id to use a test point. When the line is does notpass through the origin, I use (0, 0) for the test point.
y ≥ 2x + 1⇒ 0 ≥ 2(0) + 1⇒ 0 ≥ 1
This is a false statement. So, we shade the side that contains the testpoint.
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feasible set
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One Inequality
Then, we decide which side of the line satisfies the inequality. Theeasiest way I know id to use a test point. When the line is does notpass through the origin, I use (0, 0) for the test point.
y ≥ 2x + 1⇒ 0 ≥ 2(0) + 1⇒ 0 ≥ 1
This is a false statement. So, we shade the side that contains the testpoint.
-3 -2 -1 321-1
-2
-3
1
2
3
feasible set
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One Inequality
Then, we decide which side of the line satisfies the inequality. Theeasiest way I know id to use a test point. When the line is does notpass through the origin, I use (0, 0) for the test point.
y ≥ 2x + 1⇒ 0 ≥ 2(0) + 1⇒ 0 ≥ 1
This is a false statement. So, we shade the side that contains the testpoint.
-3 -2 -1 321-1
-2
-3
1
2
3
feasible set
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One Inequality
Then, we decide which side of the line satisfies the inequality. Theeasiest way I know id to use a test point. When the line is does notpass through the origin, I use (0, 0) for the test point.
y ≥ 2x + 1⇒ 0 ≥ 2(0) + 1⇒ 0 ≥ 1
This is a false statement. So, we shade the side that contains the testpoint.
-3 -2 -1 321-1
-2
-3
1
2
3
feasible set
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One Inequality
Then, we decide which side of the line satisfies the inequality. Theeasiest way I know id to use a test point. When the line is does notpass through the origin, I use (0, 0) for the test point.
y ≥ 2x + 1⇒ 0 ≥ 2(0) + 1⇒ 0 ≥ 1
This is a false statement. So, we shade the side that contains the testpoint.
-3 -2 -1 321-1
-2
-3
1
2
3
feasible set
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Two Inequalities
ExampleGraph the given system of inequalities.{
y ≤ -x+3y ≤ 2x
No difference here - we do the same thing as we just did, just twice.
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Two Inequalities
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0 ≤ −(0) + 3⇒ 0 ≤ 3
This is a true statement.
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Two Inequalities
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0 ≤ −(0) + 3⇒ 0 ≤ 3
This is a true statement.
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Two Inequalities
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0 ≤ −(0) + 3⇒ 0 ≤ 3
This is a true statement.
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Two Inequalities
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3
The line passes through the origin, so we need to use another point asthe test point. Why can’t we use the origin?
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Two Inequalities
-3 -2 -1 321-1
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1
2
3
The line passes through the origin, so we need to use another point asthe test point. Why can’t we use the origin?
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Two Inequalities
-3 -2 -1 321-1
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1
2
3
The line passes through the origin, so we need to use another point asthe test point. Why can’t we use the origin?
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Two Inequalities
-3 -2 -1 321-1
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Using the point (1,0), we have that 0 ≤ 2(1)⇒ 0 ≤ 2, which is true.
Which side do we shade?
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Two Inequalities
-3 -2 -1 321-1
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1
2
3
Using the point (1,0), we have that 0 ≤ 2(1)⇒ 0 ≤ 2, which is true.
Which side do we shade?
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Two Inequalities
-3 -2 -1 321-1
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1
2
3
Using the point (1,0), we have that 0 ≤ 2(1)⇒ 0 ≤ 2, which is true.
Which side do we shade?
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Two Inequalities
-3 -2 -1 321-1
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1
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3
feasible set
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Two Inequalities
-3 -2 -1 321-1
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1
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feasible set
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Another Two Inequality Example
ExampleGraph the given system of inequalities.{
y ≤ x+2y ≥ 3x-1
Here, we will test both first. Since neither line has the origin as it’sy-intercept, we can (0, 0) as test point for each.
y ≤ x + 2⇒ 0 ≤ 2 true
y ≥ 3x− 1⇒ 0 ≥ −1 true
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Another Two Inequality Example
ExampleGraph the given system of inequalities.{
y ≤ x+2y ≥ 3x-1
Here, we will test both first. Since neither line has the origin as it’sy-intercept, we can (0, 0) as test point for each.
y ≤ x + 2⇒ 0 ≤ 2 true
y ≥ 3x− 1⇒ 0 ≥ −1 true
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Another Two Inequality Example
ExampleGraph the given system of inequalities.{
y ≤ x+2y ≥ 3x-1
Here, we will test both first. Since neither line has the origin as it’sy-intercept, we can (0, 0) as test point for each.
y ≤ x + 2⇒ 0 ≤ 2
true
y ≥ 3x− 1⇒ 0 ≥ −1 true
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Another Two Inequality Example
ExampleGraph the given system of inequalities.{
y ≤ x+2y ≥ 3x-1
Here, we will test both first. Since neither line has the origin as it’sy-intercept, we can (0, 0) as test point for each.
y ≤ x + 2⇒ 0 ≤ 2 true
y ≥ 3x− 1⇒ 0 ≥ −1 true
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Another Two Inequality Example
ExampleGraph the given system of inequalities.{
y ≤ x+2y ≥ 3x-1
Here, we will test both first. Since neither line has the origin as it’sy-intercept, we can (0, 0) as test point for each.
y ≤ x + 2⇒ 0 ≤ 2 true
y ≥ 3x− 1⇒ 0 ≥ −1
true
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Another Two Inequality Example
ExampleGraph the given system of inequalities.{
y ≤ x+2y ≥ 3x-1
Here, we will test both first. Since neither line has the origin as it’sy-intercept, we can (0, 0) as test point for each.
y ≤ x + 2⇒ 0 ≤ 2 true
y ≥ 3x− 1⇒ 0 ≥ −1 true
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Another Two Inequality Example
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feasible set
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Another Two Inequality Example
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1
2
3
feasible set
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Another Two Inequality Example
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1
2
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feasible set
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Another Two Inequality Example
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1
2
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feasible set
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Another Two Inequality Example
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1
2
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feasible set
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Three Inequalities
ExampleGraph the given system of inequalities.
x + y ≤ 4x− y ≤ 5y ≤ 2x− 4
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Three Inequalities
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feasible set
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Three Inequalities
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2
4
6
feasible set
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Three Inequalities
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2
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feasible set
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Three Inequalities
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2
4
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feasible set
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Three Inequalities
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2
4
6
feasible set
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Three Inequalities
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-4
-6
2
4
6
feasible set
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Three Inequalities
-6 -4 -2 642-2
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-6
2
4
6
feasible set
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Three Inequalities
ExampleGraph the given system of inequalities.
3x− 2y ≥ 6x + y ≤ 5y ≥ −2
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Three Inequalities
-6 -4 -2 642-2
-4
-6
2
4
6
feasible set
![Page 45: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/45.jpg)
Three Inequalities
-6 -4 -2 642-2
-4
-6
2
4
6
feasible set
![Page 46: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/46.jpg)
Three Inequalities
-6 -4 -2 642-2
-4
-6
2
4
6
feasible set
![Page 47: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/47.jpg)
Three Inequalities
-6 -4 -2 642-2
-4
-6
2
4
6
feasible set
![Page 48: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/48.jpg)
Three Inequalities
-6 -4 -2 642-2
-4
-6
2
4
6
feasible set
![Page 49: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/49.jpg)
Three Inequalities
-6 -4 -2 642-2
-4
-6
2
4
6
feasible set
![Page 50: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/50.jpg)
Three Inequalities
-6 -4 -2 642-2
-4
-6
2
4
6
feasible set
![Page 51: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/51.jpg)
Multiple Inequalities
ExampleGraph the given system of inequalities.
−2 ≤ x ≤ 3−1 ≤ y ≤ 52x + y ≤ 6
![Page 52: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/52.jpg)
Multiple Inequalities
-6 -4 -2 642-2
-4
-6
2
4
6
feasible set
![Page 53: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/53.jpg)
Multiple Inequalities
-6 -4 -2 642-2
-4
-6
2
4
6
feasible set
![Page 54: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/54.jpg)
Multiple Inequalities
-6 -4 -2 642-2
-4
-6
2
4
6
feasible set
![Page 55: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/55.jpg)
Multiple Inequalities
-6 -4 -2 642-2
-4
-6
2
4
6
feasible set
![Page 56: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/56.jpg)
Multiple Inequalities
-6 -4 -2 642-2
-4
-6
2
4
6
feasible set
![Page 57: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/57.jpg)
Multiple Inequalities
-6 -4 -2 642-2
-4
-6
2
4
6
feasible set
![Page 58: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/58.jpg)
Multiple Inequalities
-6 -4 -2 642-2
-4
-6
2
4
6
feasible set
![Page 59: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/59.jpg)
Multiple Inequalities
ExampleGraph the given system of inequalities.
y ≤ x + 4y ≤ 3− xx ≥ 0, y ≥ 0
![Page 60: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/60.jpg)
Multiple Inequalities
-6 -4 -2 642-2
-4
-6
2
4
6
f.s.
![Page 61: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/61.jpg)
Multiple Inequalities
-6 -4 -2 642-2
-4
-6
2
4
6
f.s.
![Page 62: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/62.jpg)
Multiple Inequalities
-6 -4 -2 642-2
-4
-6
2
4
6
f.s.
![Page 63: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/63.jpg)
Multiple Inequalities
-6 -4 -2 642-2
-4
-6
2
4
6
f.s.
![Page 64: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/64.jpg)
Multiple Inequalities
-6 -4 -2 642-2
-4
-6
2
4
6
f.s.
![Page 65: 2.4 Graphing Inequalitiesbtravers.weebly.com › uploads › 6 › 7 › 2 › 9 › 6729909 › 2.4_graphing_i… · The difference between how you would have learned these before](https://reader034.vdocument.in/reader034/viewer/2022042406/5f1fb91e4d05f0414058aaa2/html5/thumbnails/65.jpg)
Multiple Inequalities
-6 -4 -2 642-2
-4
-6
2
4
6
f.s.