1. Warm Up Find the intercepts of each line. 1. 3x + 2y = 18
Write the function in slope-intercept form. Then graph.
4. 2x + 3y = –3
(0, 9), (6, 0)
(0, –8), (2, 0)
(0, 5), (–2, 0)

2. Linear Inequalities in Two Variables 2-5
Holt Algebra 2

3. Objectives Can you graph linear inequalities on the coordinate plane?
Can you solve problems using linear inequalities?

4. Linear functions form the basis of linear inequalities
Linear functions form the basis of linear inequalities. A linear inequality in two variables relates two variables using an inequality symbol, such as y > 2x – 4. Its graph is a region of the coordinate plane bounded by a line. The line is a boundary line, which divides the coordinate plane into two regions.

5. To graph y ≥ 2x – 4, make the boundary line solid, and shade the region above the line. To graph y > 2x – 4, make the boundary line dashed because y-values equal to 2x – 4 are not included.

6. Example 1
Graph the inequality
The boundary line is which has a y-intercept of 2 and a slope of .
Draw the boundary line dashed because it is not part of the solution.
Then shade the region above the boundary line to show .

7. Graph the inequality y ≤ –1.
Example 2
Graph the inequality y ≤ –1.
Recall that y= –1 is a horizontal line.
Step 1 Draw a solid line for y=–1 because the boundary line is part of the graph.
Step 2 Shade the region below the boundary line to show where y < –1. .

8. Example 3
Graph the inequality y ≥ 3x –2.
The boundary line is y = 3x – 2 which has a
y–intercept of –2 and a slope of 3.
Draw a solid line because it is part of the solution.
Then shade the region above the boundary line to show y > 3x – 2.

9. Graph the inequality y < –3.
Example 4
Graph the inequality y < –3.
Recall that y = –3 is a horizontal line.
Step 1 Draw the boundary line dashed because it is not part of the solution.
Step 2 Shade the region below the boundary line to show where y < –3. .

10. Example 5
Graph 3x + 4y ≤ 12 using intercepts.
Step 1 Find the intercepts.
Substitute x = 0 and y = 0 into 3x + 4y = 12 to find the intercepts of the boundary line.
y-intercept
x-intercept
3x + 4y = 12
3x + 4y = 12
3(0) + 4y = 12
3x + 4(0) = 12
4y = 12
3x = 12
y = 3
x = 4

11. Step 2 Draw the boundary line.
Example 5 Continued
Step 2 Draw the boundary line.
The line goes through (0, 3) and (4, 0). Draw a solid line for the boundary line because it is part of the graph.
(0, 3)
Step 3 Find the correct region to shade.
Substitute (0, 0) into the inequality. Because ≤ 12 is true, shade the region that contains (0, 0).
(4, 0)

12. Example 6
Graph 3x – 4y > 12 using intercepts.
Step 1 Find the intercepts.
Substitute x = 0 and y = 0 into 3x – 4y = 12 to find the intercepts of the boundary line.
y-intercept
x-intercept
3x – 4y = 12
3x – 4y = 12
3(0) – 4y = 12
3x – 4(0) = 12
– 4y = 12
3x = 12
y = – 3
x = 4

13. Example 7: Problem-Solving Application
A school carnival charges $4.50 for adults and $3.00 for children. The school needs to make at least $135 to cover expenses.
A. Using x as the adult ticket price and y as the child ticket price, write and graph an inequality for the amount the school makes on ticket sales.
B. If 25 child tickets are sold, how many adult tickets must be sold to cover expenses?

14. An inequality that models the problem is 4.5x + 3y ≥ 135.2
Make a Plan
Let x represent the number of adult tickets and y represent the number of child tickets that must be sold. Write an inequality to represent the situation.
135 y
3.00 + x
4.50
total.
is at least
number of child tickets
times
child price
plus
number of adult tickets
Adult price •
An inequality that models the problem is 4.5x + 3y ≥ 135.

15. Solve 3
Find the intercepts of the boundary line.
4.5(0) + 3y = 135
4.5x + 3(0) = 135
y = 45
x = 30
Graph the boundary line through (0, 45) and (30, 0) as a solid line.
Shade the region above the line that is in the first quadrant, as ticket sales cannot be negative.

16. If 25 child tickets are sold,
4.5x + 3(25) ≥ 135
Substitute 25 for y in 4.5x + 3y ≥ 135.
4.5x + 75 ≥ 135
Multiply 3 by 25.
4.5x ≥ 60, so x ≥ 13.3 _
A whole number of tickets must be sold.
At least 14 adult tickets must be sold.
14($4.50) + 25($3.00) = $138.00, so the answer is reasonable.

17. Example 8
A café gives away prizes. A large prize costs the café $125, and the small prize costs $40. The café will not spend more than $1500. How many of each prize can be awarded? How many small prizes can be awarded if 4 large prizes are given away?

18. Solve 3
Find the intercepts of the boundary line.
40(0) + 125y = 1500
40x + 125(0) = 1500
y = 12
x = 37.5
Graph the boundary line through (0, 12) and (37.5, 0) as a solid line.
Shade the region below the line that is in the first quadrant, as prizes awarded cannot be negative.

19. Bellwork 2.5
1. Graph 2x –5y  10 using intercepts.
2. Solve –6y < 18x – 12 for y. Graph the solution.
y > –3x + 2

20. Bellwork 2.5
3. Potatoes cost a chef $18 a box, and carrots cost $12 a box. The chef wants to spend no more than $144. Use x as the number of boxes of potatoes and y as the number of boxes of carrots.
a. Write an inequality for the number of boxes the chef can buy.
18x + 12y ≤ 144
b. How many boxes of potatoes can the chef order if she orders 4 boxes of carrot?
no more than 5