1. A4.e
How Do I Graph The Solution Set of A Linear Inequality in Two Variables?
Course 3
Warm Up
Problem of the Day
Lesson Presentation

2. Graphing Inequalities in Two Variables
Course 3
12-6
Graphing Inequalities in Two Variables
Warm Up
Find each equation of direct variation, given that y varies directly with x.
1. y is 18 when x is 3.
2. x is 60 when y is 12.
3. y is 126 when x is 18.
4. x is 4 when y is 20.
y = 6x
y = x 1 5
y = 7x
y = 5x

3. Graphing Inequalities in Two Variables
Course 3
12-6
Graphing Inequalities in Two Variables
Problem of the Day
The circumference of a pizza varies directly with its diameter. If you graph that direct variation, what will the slope be? 

4. Learn to graph inequalities on the coordinate plane.
Course 3
12-6
Graphing Inequalities in Two Variables
Learn to graph inequalities on the coordinate plane.

5. Insert Lesson Title Here
Course 3
12-6
Graphing Inequalities in Two Variables
Insert Lesson Title Here
Vocabulary
boundary line
linear inequality

6. Graphing Inequalities in Two Variables
Course 3
12-6
Graphing Inequalities in Two Variables
A graph of a linear equation separates the coordinate plane into three parts: the points on one side of the line, the points on the boundary line, and the points on the other side of the line.

7. Graphing Inequalities in Two Variables
Course 3
12-6
Graphing Inequalities in Two Variables

8. Graphing Inequalities in Two Variables
Course 3
12-6
Graphing Inequalities in Two Variables
When the equality symbol is replaced in a linear equation by an inequality symbol, the statement is a linear inequality. Any ordered pair that makes the linear inequality true is a solution.

9. Example 1A: Graphing Inequalities
Course 3
12-6
Graphing Inequalities in Two Variables
Example 1A: Graphing Inequalities
Graph each inequality.
y < x – 1
First graph the boundary line y = x – 1. Since no points that are on the line are solutions of y < x – 1, make the line dashed (creates 2 open half planes). Then determine on which side of the line the solutions lie.
(0, 0)
Test a point not on the line.
y < x – 1
0 < 0 – 1 ?
Substitute 0 for x and 0 for y.
0 < –1 ?

10. Graphing Inequalities in Two Variables
Course 3
12-6
Graphing Inequalities in Two Variables
Any point on the line y = x -1 is not a solution of y < x - 1 because the inequality symbol < means only “less than” and does not include “equal to.”
Helpful Hint

11. Graphing Inequalities in Two Variables
Course 3
12-6
Graphing Inequalities in Two Variables
Example 1A Continued
Since 0 < –1 is not true, (0, 0) is not a solution of y < x – 1. Shade the side of the line that does not include (0, 0). When a “<“ is used, shade below the boundary line.
(0, 0)

12. Additional Example 1B: Graphing Inequalities
Course 3
12-6
Graphing Inequalities in Two Variables
Additional Example 1B: Graphing Inequalities
y  2x + 1
Graph the boundary line: y = 2x + 1.
2. Make the line solid (creates 2 closed half planes) because the points that are on the line are solutions of y  2x + 1.
3. Shade above the line (because the  is used). This is where the rest of the solutions of y  2x + 1 lie.
(0, 4)
Choose any point not on the line.
y ≥ 2x + 1
4 ≥ 0 + 1 ?
Substitute 0 for x and 4 for y.

13. Graphing Inequalities in Two Variables
Course 3
12-6
Graphing Inequalities in Two Variables
Any point on the line y = 2x + 1 is a solution of y ≥ 2x + 1 because the inequality symbol ≥ means “greater than or equal to.”
Helpful Hint

14. Additional Example 1B Continued
Course 3
12-6
Graphing Inequalities in Two Variables
Additional Example 1B Continued
Since 4  1 is true, (0, 4) is a solution of y  2x + 1. Shade the side of the line that includes (0, 4). When a “>” is used, shade above the boundary line.
(0, 4)

15. Additional Example 1C: Graphing Inequalities
Course 3
12-6
Graphing Inequalities in Two Variables
Additional Example 1C: Graphing Inequalities
2y + 5x < 6
First write the equation in slope-intercept form.
2y + 5x < 6
2y < –5x + 6
Subtract 5x from both sides.
y < – x + 3 5 2
Divide both sides by 2.
Then graph the line y = – x + 3. Since points that are on the line are not solutions of y < – x + 3, make the line dashed. Then determine on which side of the line the solutions lie. 5 2

16. Additional Example 1C Continued
Course 3
12-6
Graphing Inequalities in Two Variables
Additional Example 1C Continued
(0, 0)
Choose any point not on the line.
y < – x + 3 5 2
0 < 0 + 3 ?
Substitute 0 for x and 0 for y.
0 < 3 ?
Since 0 < 3 is true, (0, 0) is a solution of y < – x + 3. Shade the side of the line that includes (0, 0). 5 2
(0, 0)

17. Graphing Inequalities in Two Variables
Course 3
12-6
Graphing Inequalities in Two Variables
Check It Out: Example 1A
Graph each inequality.
y < x – 4
First graph the boundary line y = x – 4. Since no points that are on the line are solutions of y < x – 4, make the line dashed. Then determine on which side of the line the solutions lie.
(0, 0)
Test a point not on the line.
y < x – 4
0 < 0 – 4 ?
Substitute 0 for x and 0 for y.
0 < –4 ?

18. Check It Out: Example 1A Continued
Course 3
12-6
Graphing Inequalities in Two Variables
Check It Out: Example 1A Continued
Since 0 < –4 is not true, (0, 0) is not a solution of y < x – 4. Shade the side of the line that does not include (0, 0).
(0, 0)

19. Graphing Inequalities in Two Variables
Course 3
12-6
Graphing Inequalities in Two Variables
Check It Out: Example 1B
y > 4x + 4
First graph the boundary line y = 4x + 4. Since points that are on the line are solutions of
y  4x + 4, make the line solid. Then shade the part of the coordinate plane in which the rest of the solutions of y  4x + 4 lie.
(2, 3)
Choose any point not on the line.
y ≥ 4x + 4
3 ≥ 8 + 4 ?
Substitute 2 for x and 3 for y.

20. Check It Out: Example 1B Continued
Course 3
12-6
Graphing Inequalities in Two Variables
Check It Out: Example 1B Continued
Since 3  12 is not true, (2, 3) is not a solution of y  4x + 4. Shade the side of the line that does not include (2, 3).
(2, 3)

21. Graphing Inequalities in Two Variables
Course 3
12-6
Graphing Inequalities in Two Variables
Check It Out: Example 1C
3y + 4x  9
First write the equation in slope-intercept form.
3y + 4x  9
3y  –4x + 9
Subtract 4x from both sides.
y  – x + 3 4 3
Divide both sides by 3. 4 3
Then graph the line y = – x + 3. Since points that are on the line are solutions of y  – x + 3, make the line solid. Then determine on which side of the line the solutions lie.

22. Check It Out: Example 1C Continued
Course 3
12-6
Graphing Inequalities in Two Variables
Check It Out: Example 1C Continued
(0, 0)
Choose any point not on the line.
y  – x + 3 4 3
(0, 0)
0  0 + 3 ?
Substitute 0 for x and 0 for y.
0  3 ?
Since 0  3 is not true, (0, 0) is not a solution of y  – x + 3. Shade the side of the line that does not include (0, 0). 4 3

23. Additional Example 2: Career Application
Course 3
12-6
Graphing Inequalities in Two Variables
Additional Example 2: Career Application
A successful screenwriter can write no more than seven and a half pages of dialogue each day. Graph the relationship between the number of pages the writer can write and the number of days. At this rate, would the writer be able to write a 200-page screenplay in 30 days?
First find the equation of the line that corresponds to the inequality.
In 0 days the writer writes 0 pages.
point (0, 0)
In 1 day the writer writes no more than 7 pages. 1 2
point (1, 7.5)

24. Graphing Inequalities in Two Variables
Course 3
12-6
Graphing Inequalities in Two Variables
The phrase “no more” can be translated as less than or equal to.
Helpful Hint

25. Additional Example 2 Continued
Course 3
12-6
Graphing Inequalities in Two Variables
Additional Example 2 Continued
With two known points, find the slope.
m =
7.5 – 0
1 – 0
7.5 1 =
= 7.5
y  7.5 x + 0
The y-intercept is 0.
Graph the boundary line y = 7.5x. Since points on the line are solutions of y  7.5x make the line solid. Shade the part of the coordinate plane in which the rest of the solutions of y  7.5x lie.

26. Additional Example 2 Continued
Course 3
12-6
Graphing Inequalities in Two Variables
Additional Example 2 Continued
(2, 2) Choose any point not on the line.
y  7.5x
2  7.5  2 ?
Substitute 2 for x and 2 for y.
2  15  ?
Since 2  15 is true, (2, 2) is a solution of y  7.5x. Shade the side of the line that includes point (2, 2).

27. Additional Example 2 Continued
Course 3
12-6
Graphing Inequalities in Two Variables
Additional Example 2 Continued
The point (30, 200) is included in the shaded area, so the writer should be able to complete the 200 page screenplay in 30 days.

28. Graphing Inequalities in Two Variables
Course 3
12-6
Graphing Inequalities in Two Variables
Check It Out: Example 2
A certain author can write no more than 20 pages every 5 days. Graph the relationship between the number of pages the writer can write and the number of days. At this rate, would the writer be able to write 140 pages in 20 days?
First find the equation of the line that corresponds to the inequality.
In 0 days the writer writes 0 pages.
point (0, 0)
In 5 days the writer writes no more than 20 pages.
point (5, 20)

29. Check It Out: Example 2 Continued
Course 3
12-6
Graphing Inequalities in Two Variables
Check It Out: Example 2 Continued
20 - 0
5 - 0
m = = 20 5
= 4
With two known points, find the slope.
The y-intercept is 0.
y  4x + 0
Graph the boundary line y = 4x. Since points on the line are solutions of y  4x make the line solid. Shade the part of the coordinate plane in which the rest of the solutions of y  4x lie.

30. Check It Out: Example 2 Continued
Course 3
12-6
Graphing Inequalities in Two Variables
Check It Out: Example 2 Continued
(5, 60) Choose any point not on the line.
y  4x
60  4  5 ?
Substitute 5 for x and 60 for y.
60  20  ?
Since 60  20 is not true, (5, 60) is not a solution of y  4x. Shade the side of the line that does not include (5, 60).

31. Check It Out: Example 2 Continued
Course 3
12-6
Graphing Inequalities in Two Variables
Check It Out: Example 2 Continued y
200
180
160
140
120
100 80 60
40\ 20
Pages x
Days
The point (20, 140) is not included in the shaded area, so the writer will not be able to write 140 pages in 20 days.

32. Graphing Inequalities in Two Variables
Course 3
12-6
Graphing Inequalities in Two Variables
Lesson Quiz Part I
Graph each inequality.
1. y < – x + 4 1 3

33. Graphing Inequalities in Two Variables
Course 3
12-6
Graphing Inequalities in Two Variables
Lesson Quiz Part II
2. 4y + 2x > 12

34. Graphing Inequalities in Two Variables
Course 3
12-6
Graphing Inequalities in Two Variables
Lesson Quiz: Part III
Tell whether the given ordered pair is a solution of each inequality.
3. y < x + 15 (–2, 8)
4. y  3x – 1 (7, –1)
yes no