§ 2.4 Graphing Inequalities

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.

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.

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.

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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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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-3

1

2

3

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

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

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

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

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

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.

Two Inequalities

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0 ≤ −(0) + 3⇒ 0 ≤ 3

This is a true statement.

Two Inequalities

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0 ≤ −(0) + 3⇒ 0 ≤ 3

This is a true statement.

Two Inequalities

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0 ≤ −(0) + 3⇒ 0 ≤ 3

This is a true statement.

Two Inequalities

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The line passes through the origin, so we need to use another point asthe test point. Why can’t we use the origin?

Two Inequalities

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The line passes through the origin, so we need to use another point asthe test point. Why can’t we use the origin?

Two Inequalities

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The line passes through the origin, so we need to use another point asthe test point. Why can’t we use the origin?

Two Inequalities

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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?

Two Inequalities

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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?

Two Inequalities

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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?

Two Inequalities

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1

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3

feasible set

Two Inequalities

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feasible set

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

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

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

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

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

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

Another Two Inequality Example

-3 -2 -1 321-1

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1

2

3

feasible set

Another Two Inequality Example

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-3

1

2

3

feasible set

Another Two Inequality Example

-3 -2 -1 321-1

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1

2

3

feasible set

Another Two Inequality Example

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1

2

3

feasible set

Another Two Inequality Example

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1

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3

feasible set

Three Inequalities

ExampleGraph the given system of inequalities.

x + y ≤ 4x− y ≤ 5y ≤ 2x− 4

Three Inequalities

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feasible set

Three Inequalities

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6

feasible set

Three Inequalities

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feasible set

Three Inequalities

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feasible set

Three Inequalities

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feasible set

Three Inequalities

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feasible set

Three Inequalities

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feasible set

Three Inequalities

ExampleGraph the given system of inequalities.

3x− 2y ≥ 6x + y ≤ 5y ≥ −2

Three Inequalities

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2

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6

feasible set

Three Inequalities

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feasible set

Three Inequalities

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feasible set

Three Inequalities

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feasible set

Three Inequalities

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feasible set

Three Inequalities

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feasible set

Three Inequalities

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feasible set

Multiple Inequalities

ExampleGraph the given system of inequalities.

−2 ≤ x ≤ 3−1 ≤ y ≤ 52x + y ≤ 6

Multiple Inequalities

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feasible set

Multiple Inequalities

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feasible set

Multiple Inequalities

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feasible set

Multiple Inequalities

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feasible set

Multiple Inequalities

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feasible set

Multiple Inequalities

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6

feasible set

Multiple Inequalities

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feasible set

Multiple Inequalities

ExampleGraph the given system of inequalities.

y ≤ x + 4y ≤ 3− xx ≥ 0, y ≥ 0

Multiple Inequalities

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f.s.

Multiple Inequalities

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f.s.

Multiple Inequalities

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f.s.

Multiple Inequalities

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f.s.

Multiple Inequalities

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f.s.

Multiple Inequalities

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f.s.