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MAT 085 – Intermediate Algebra-Part I Section 9.4 (Bittinger CA7)

[email protected] kradermath.jimdo.com 09/2017

Topics to be Covered and Student Learning Outcomes (SLOs) After we cover Section 9.4, you should be able to:

1. Describe what it means to solve an inequality. [SLO 1] 2. Use the graphical method to solve a linear inequality. [SLO 4] 3. Describe what is meant by a system of inequalities. [SLO 1] 4. Describe what it means to solve a system of inequalities. [SLO 1] 5. Use the graphical method to solve a system of linear inequalities. [SLO 4]

Graphing Linear Equations and Linear Inequalities in Two Variables In Chapter 3, we learned how to graph linear equations in two variables, e.g.:

2 1y x .

Now we will learn how to graph linear inequalities in two variables, e.g.:

2 1

2 1

2 1

2 1

y x

y x

y x

y x

The graph of a linear inequality shows all of the solutions of the inequality, just like the graph of a linear equation shows all of the solutions of the equation. EXAMPLE: Graphing Linear Equations and Linear Inequalities in Two Variables

Slide 5GHK 09/2017MAT 085 (Bittinger CA7) – 9.4

The y-coordinate increases as we move up. Therefore, y>2x+1 for every ordered pair that lies above the line.

The y-coordinate decreases as we move down. Therefore, y<2x+1 for every ordered pair that lies below the line.

y=2x+1 for every ordered pair that lies on the line.

y increases

y decreases



Graphing Linear Equations and Linear Inequalities in Two Variables Graph of a linear equation in two variables:

A picture that shows all of the solutions of the equation, i.e., all ordered pairs

,x y that make the equation true.

The graph of a linear equation in two variables is a straight line. Graph of a linear inequality in two variables:

A picture that shows all of the solutions of the inequality, i.e., all ordered pairs

,x y that make the inequality true.

The graph of a linear inequality in two variables is a half-plane. EXAMPLES: Graphing Linear Inequalities in Two Variables The shaded area is the graph, i.e., it represents the solutions of the inequality.

2 1y x 2 1y x

A dashed line indicates that the ordered pairs on the line are not part of the solution.

2 1y x 2 1y x

A solid line indicates that the ordered pairs on the line are part of the solution.



EXAMPLES: Graphing Linear Inequalities in Two Variables (cont’d) Use substitution to determine whether the following ordered pairs are solutions of the

inequality 2 1y x .

2 1y x 1,7

1,3

1, 5

EXAMPLES: Graphing Linear Inequalities in Two Variables (cont’d) Graphs of inequalities can also involve horizontal or vertical lines.

7y 4x

Slide 9

Use substitution to determine whether an ordered pair is a solution of an inequality

GHK 09/2017MAT 085 (Bittinger CA7) – 9.4

2 1y x



EXAMPLES: Graphing Linear Inequalities in Two Variables (cont’d) Graph the following linear inequalities:

2 5 15x y

3 2 6x y

x

y

x

y



Solving and Graphing Inequalities METHOD 1:

If necessary, solve the inequality for y. o Remember to reverse the inequality symbol whenever you multiply or

divide both sides of the inequality by a negative number! Graph the line that corresponds to the given equation.

o Draw a solid line for inequalities using or . o Draw a dashed line for inequalities using or .

Use the inequality symbol to determine whether to shade above or below the line. o Shade above the line for inequalities of the form y mx b or y mx b .

o Shade below the line for inequalities of the form y mx b or y mx b .

Use a test point that does not lie on the line to check your graph.

o 0,0 is a convenient test point unless the line passes through the origin.

METHOD 2 (Especially useful if you need to solve for y):

Write the corresponding equation in slope-intercept form by solving for y (i.e., forget about the inequality symbol for the moment).

Graph the line that corresponds to the given equation. o Draw a solid line for inequalities using or . o Draw a dashed line for inequalities using or .

Use a test point that does not lie on the line to determine whether to shade above or below the line.

o If the test point is a solution of the inequality, shade the half-plane containing the test point.

o If the test point is not a solution, shade the other half-plane.

o 0,0 is a convenient test point unless the line passes through the origin.

EXAMPLES: Graphing Linear Inequalities in Two Variables Graph the following linear inequality using METHOD 2.

3 2 6x y

Based on the previous example, we see that the equation 3 2 6x y can be

written in slope-intercept form as

32

3y x . (Draw a dashed line because

of the inequality symbol.) Now pick a test point to determine

whether to shade above or below the line.

Slide 16MAT 085 (Bittinger CA7) – 9.4 GHK 09/2017



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Systems of Linear Inequalities

NOTE: The concept of a “solution” is the same in both cases: the set of ordered pairs that make both equations – or both inequalities – true. EXAMPLES: Graphing Systems of Linear Inequalities Graph the following system of linear inequalities:

3 0

2 5 15

x y

x y

How many solutions does the system of linear inequalities have?

Slide 18

Systems of linear inequalities

System of two linear equations

The solution of a system of equations is the set of ordered pairs that makes both equationstrue.

Algebraic methods (the substitution and elimination methods) are the most reliable ways to solve systems of linear equations.

System of two linear inequalities

The solution of a system of inequalities is the set of ordered pair that makes both inequalitiestrue.

Graphical methods are used to solve systems of linear inequalities. Graph each inequality separately.

Find ordered pairs that lie in bothsolution sets (i.e., the intersection of the two graphs).

GHK 09/2017

3 0

2 5 15

x y

x y

3 0

2 5 15

x y

x y

MAT 085 (Bittinger CA7) – 9.4

x

y



EXAMPLES: Graphing Systems of Linear Inequalities (cont’d) Graph the following systems of linear inequalities:

4 2 6

2 8

x y

x y

3 4y

x

y

x

y



EXAMPLES: Graphing Systems of Linear Inequalities (cont’d) Systems of linear inequalities can also involve vertical lines.

3 4x

3

4

x

y

EXAMPLES: Graphing Systems of Linear Inequalities (cont’d) Graph the following systems of linear inequalities:

4 5 30

5

x y

y


System of inequalities with two vertical lines

3

4

x

y

3 4x

System of inequalities with one horizontal and one vertical line


System of inequalities with two vertical lines

3

4

x

y

3 4x

System of inequalities with one horizontal and one vertical line

x

y



Summary: Solving (i.e., Graphing) Systems of Linear Inequalities in Two Variables

Graph the solution of each linear inequality separately. The solution of the system of linear inequalities consists of the ordered pairs that lie

in both solution sets, i.e., the intersection of the two graphs.

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