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Algebra 1

2.15.2 Determining if Points on Boundary Lines Are Solutions to a System

Algebra 12.15.2 Determining if Points on Boundary Lines Are Solutions to a System

2.15.2 Determining if Points on Boundary Lines Are Solutions to a System

2.15.2 • Determining if Points on Boundary Lines Are Solutions to a System

Activity

Here are the graphs of the inequalities in this system:

{ x < y y ≥ − 2 x − 6 { x < y y ≥ − 2 x − 6

Decide whether each point is a solution to the system. Be prepared to explain how you know.

1.

( 3 , − 5 ) ( 3 , − 5 )

2.

( 0 , 5 ) ( 0 , 5 )

3.

( − 6 , 6 ) ( − 6 , 6 )

4.

( 3 , 3 ) ( 3 , 3 )

5.

( − 2 , − 2 ) ( − 2 , − 2 )

Are you ready for more?

Extending Your Thinking

Find a system of inequalities with this triangle as its set of solutions:

Video: Determining if Points on Boundary Lines Are Solutions to a System

Solving Systems of Inequalities

Self Check

Here is the graph of this system of inequalities. Which of the following ordered pairs falls within the solution set?

6 x + 4 y ≤ 34 6 x + 4 y ≤ 34

− 2 x + 7 y > 22 − 2 x + 7 y > 22

(10,6) (10,6)

Additional Resources

Determine Solutions of Systems of Inequalities on Boundary Lines

A solution to a system of inequalities only exists on a boundary line if the line is solid, not dashed.

Example

Solve the system by graphing:

{ x − y > 3 y < − 1 5 x + 4 . { x − y > 3 y < − 1 5 x + 4 .

Solution

{ x − y > 3 y < − 1 5 x + 4 { x − y > 3 y < − 1 5 x + 4

Graph x − y > 3 x − y > 3 , by graphing x − y = 3 x − y = 3  and testing a point. The intercepts are x = 3 x = 3 and y = − 3 y = − 3 , and the boundary line will be dashed. Test ( 0 , 0 ) ( 0 , 0 ) , which makes the inequality false, so shade (red) the side that does not contain ( 0 , 0 ) ( 0 , 0 ) .

Graph y < − 1 5 x + 4 y < − 1 5 x + 4 by graphing y = − 1 5 x + 4 y = − 1 5 x + 4 using the slope m = − 15 m = − 15 and y y -intercept b = 4 b = 4 . The boundary line will be dashed

Test ( 0 , 0 ) ( 0 , 0 ) , which makes the inequality true, so shade (blue) the side that contains ( 0 , 0 ) ( 0 , 0 ) .

Choose a test point in the solution and verify that it is a solution to both inequalities.

The point of intersection of the two lines is not included because both boundary lines were dashed. The solution is the area shaded twice—which appears as the darkest shaded region.

Notice that the point ( 0 , − 3 ) ( 0 , − 3 ) is NOT a solution because it falls on the red dashed line. The point ( 4 , − 2 ) ( 4 , − 2 ) IS a solution because it is included in the solutions to both inequalities.

Try it

Determine Solutions of Systems of Inequalities on Boundary Lines

Which of the following are solutions of the graph below?

  • ( 2 , 1 ) ( 2 , 1 )
  • ( 2 , 3 ) ( 2 , 3 )
  • ( 5 , 0 ) ( 5 , 0 )
  • ( 3 , − 1 ) ( 3 , − 1 )
  • ( − 1 , 6 ) ( − 1 , 6 )
  • ( 1 , 6 ) ( 1 , 6 )

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