Graphing Linear Inequalities in Two Variables

10.7 Graphing Linear Inequalities in Two Variables

Learning Objectives: A. Determine whether an ordered pair is a solution of a linear inequality in two variables. B. Graph a linear inequality in two variables.

Key Vocabulary: linear inequality in two variables, halfplanes, boundary line.

Definitions:

A linear inequality in two variables can be written in the form of

Ax + By < C ;

Ax + By > C ; Ax + By C ;

Ax + By C

Steps to graph linear inequalities.

1. Replace the inequality symbol with an equals sign and graph the resulting line.

If inequality involving or , draw a solid line.

If inequality involving < or >, draw a dashed line.

2. Pick a test point that does NOT lie on the line. Substitute the values in the inequality.

If the result is true, shade the side that contain the test point.

If a false statement, shade the other side.

3. Any point (ordered pair) in the shaded region is a solution of the inequality.

CAUTION! If multiply or divide by a negative number, the inequality sign change to opposite.

Class notes:

5

Example 1.

Graph the following inequality. Label at least two points on the graph grid.

1. 2x - 4 y < - 8

?5

5

?5

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5

2. x + 2 0

?5

5

1

?5

10.7 Exercises

Determine which ordered pairs are

14.

solutions of the linear inequality

3x - 4y 2

1. (-1, -2) 2. (2, 1) 3. (3, 2)

4.

Graph each inequality and label at least two points on the graph grid.

5. x - 4 y > 4

6. y < 5x

15.

7. x -4 y

8. 8x - 2 y 16

9. x - y > 3

10. y 3x + 4

11. x + 2 > 0

12. y + 2 < 0

Use the graphs below to write the inequality. Also, find two points that are solutions of the inequality and check them.

13.

2

Some challenging questions:

1. Find the equation of the line that passes through (2.5, -1.7) if it is perpendicular to the

line

containing

the

points

1 ,

3

and

1

,-

1 .

Write

your

answer

in

the

form

2 8 4 4

Ax + By = C .

2. Let f (x) = x2 - 7x +1. Let g(x) = 5x + 3 . Find g( f (- 2)) .

3. The graph below shows the weekly average number of hours of sleep and the weekly average number of hours of TV viewing of each of five friends.

Using these clues, you need to tell which point represents which person: 1. Adam watches fewer hours of TV than his friends. 2. Corey gets at least one more hour of sleep each night than Stacey. 3. Of all of them, Stacey gets the least sleep. 4. Leland watches more TV than Manuel does. A is _______________________ B is _______________________ C is _______________________ D is _______________________ E is _______________________

3

4. Each graph shows the trips of two families.

The Li family left before the Rodriguez family. Both families took the same route. a. Which graph shows their trips? Graph _________ b. Which line represents the Li's trip? Line __________ c. How many hours was the Rodriguez's trip? __________ d. Which family stopped to shop? _______________________ How does that graph show this?

The Clarks and the Jeffersons left on their trips at the same time. They took the same route. e. The Clarks stopped for lunch. Which line represents the Clark's trip? Line ____ f. At what time did the Jeffersons complete their trip? _______________ g. Which family drove faster? ______________ How can you tell from the graph? h. Which family averaged more than 30 miles per hour?

4

5. For numbers a-e, sketch a graph that would be a reasonable model for the situation described. Label axes with appropriate quantities and their units. For number 5 write a story for the graph. a. The birthday gift he received was such a large amount of money that he used it to start a savings account to which he regularly added a fixed amount.

b. A car starts out slowly and then goes faster and faster until a tire blows out.

c. They started their business with a large no-interest loan from a rich uncle. Each time they sold one of their products they were able to pay down the loan.

d. When she took up jogging, she was very slow and she felt like it took forever. Each time she ran she was able to go a little faster. She was glad because then she had to set aside less time for her run, and had more time for her latte afterward.

e.

speed

time

5

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