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Chapter 6 190

6-6 Systems of Linear Inequalities

1. Circle the system of equations.

3x 1 2y 5 20 3x 2 2y 5 12 3x 1 2y 5 12 7x 1 2y , 10 4x 1 4y 5 20

2. How can you tell if an ordered pair is a solution to a system of equations?

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Vocabulary Builder

boundary (noun) BOWN dree

Related Words: bounds (noun or verb), border (noun or verb), bounded (adjective)

Definition: A boundary is something that divides one item from another.

Math Usage: The graph of a linear inequality in two variables is a region bounded by a line. All points on one side of the boundary line are solutions.

Use Your Vocabulary

3. Complete each sentence with bounds, boundary, or border.

The landscaper planted a 9 of flowers along the sidewalk.

The ball bounced out of 9.

The 9 line on the map between countries is shown in red.

4. Circle the ordered pair that is on the boundary of y 1 2 # x 2 5.

(9, 0) (0, 26) (6, 21) (10, 5) (5, 23)

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191 Lesson 6-6

Graphing a System of Inequalities

Got It? What is the graph of the system? y L 2x 1 5 23x 1 y K 24

5. The boundary line for y $ 2x 1 5 will be solid / dashed .

The boundary line for 23x 1 y # 24 will be solid / dashed .

6. Write the inequality 23x 1 y # 24 in slope-intercept form.

y # ? x 2 4

7. Graph the inequalities on the coordinate grid at the right.

Writing a System of Inequalities From a Graph

Got It? What system of inequalities is represented by the graph?

8. Find the inequality shown by the red line. 9. Find the inequality shown by the blue line.

Determine the equation of the red line. Determine the equation of the blue line.

y 5 y 5

The shaded region is above / below the The shaded region is above / below the

line. The boundary line is solid / dashed . line. The boundary line is solid / dashed .

Circle the inequality symbol you should use Circle the inequality symbol you should useto write the inequality for this graph. to write the inequality for this graph.

. , # $ . , # $

10. Write the system of inequalities represented by the graph.

1.)       y ≤ 2x −1                                               2.)     3x − 2y ≤ 4                     y ≥ −x + 3                                                               x + 3y ≤ 6  

         3.)    2x + y ≤ 4                                    4.)    −y ≤ 3x + 4                  −3x + y < 3                                                −3x + 3y ≤ −9     y ≥ −4    

             

 

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Lesson Check

Math Success

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193 Lesson 6-6

Check off the vocabulary words that you understand.

system of linear inequalities solution of a system of linear inequalities

Rate how well you can solve a system of linear inequalities.

• Do you UNDERSTAND?

Reasoning Suppose you are graphing a system of two linear inequalities, and the boundary lines for the inequalities are parallel. Does that mean that the system has no solution? Explain.

14. Draw a line from each graph to the corresponding system of linear inequalities.

y # 234x 1 6 y $ 23

4x 1 6 y $ 234x 1 6

y $ 234x 1 3 y # 23

4x 1 3 y $ 234x 1 3

15. Which of the systems have solutions?

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16. Would the system of y # 234x 1 6 and y # 23

4x 1 3 have a solution? If so, describe the solution.

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17. Is it possible for a system of two linear inequalities with parallel boundary lines to have a solution? Yes / No

18. Is it possible for a system of two linear inequalities with parallel boundary lines to have no solution? Yes / No

 

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Problem 3

Define what each variable represents.

Let x 5 the width of the dog run.

Let y 5 .

Graph the system of inequalities.

Write a system of inequalities.

2x 1 2y

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In slope-intercept form, this is y .

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Chapter 6 192

Using a System of Inequalities

Got It? You want to build a fence for a rectangular dog run. You want the run to be at least 10 ft wide. The run can be at most 50 ft long. You have 126 ft of fencing. What is a graph showing the possible dimensions of the dog run?

Remember, a dog run must have fencing on all four sides.

11. Follow the steps to write a system of inequalities and graph it.

12. Why is the graph only in the first quadrant?

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13. Use the graph to write possible dimensions of the dog run.

width (x): width (x): width (x):

length (y): length (y): length (y):