1. Solving Systems of Linear Inequalities Warm Up Lesson Presentation
Lesson Quiz
Holt McDougal Algebra 1
Holt Algebra 1

2. A system of linear inequalities is a set of two or more linear inequalities containing two or more variables. The solutions of a system of linear inequalities are all the ordered pairs that satisfy all the linear inequalities in the system.

3. Example 1A: Identifying Solutions of Systems of Linear Inequalities
Tell whether the ordered pair is a solution of the given system.
y ≤ –3x + 1
(–1, –3);
y < 2x + 2
(–1, –3)
(–1, –3)
y ≤ –3x + 1
y < 2x + 2
–3 –3(–1) + 1 – – ≤ 
–3 –2 + 2 –
< 
– (–1) + 2
(–1, –3) is a solution to the system because it satisfies both inequalities.

4. Example 1B: Identifying Solutions of Systems of Linear Inequalities
Tell whether the ordered pair is a solution of the given system.
y < –2x – 1
(–1, 5);
y ≥ x + 3
(–1, 5)
(–1, 5)
y < –2x – 1 ≥ 
5 –1 + 3
y ≥ x + 3
5 –2(–1) – 1
– 1
< 
(–1, 5) is not a solution to the system because it does not satisfy both inequalities.

5.   Check It Out! Example 1a
Tell whether the ordered pair is a solution of the given system.
y < –3x + 2
(0, 1);
y ≥ x – 1
(0, 1)
(0, 1)
y < –3x + 2
y ≥ x – 1
–3(0) + 2
<
1 –1 ≥ 
– 1 
(0, 1) is a solution to the system because it satisfies both inequalities.

6.   Check It Out! Example 1b
Tell whether the ordered pair is a solution of the given system.
y > –x + 1
(0, 0);
y > x – 1
(0, 0)
(0, 0)
y > –x + 1 –1 ≥ 
– 1
y > x – 1
0 –1(0) + 1
> 
(0, 0) is not a solution to the system because it does not satisfy both inequalities.

7. To show all the solutions of a system of linear inequalities, graph the solutions of each inequality. The solutions of the system are represented by the overlapping shaded regions. Below are graphs of Examples 1A and 1B on p. 435.

8. Example 2A: Solving a System of Linear Inequalities by Graphing
Graph the system of linear inequalities. Give two ordered pairs that are solutions and two that are not solutions.
(–1, 4)
(2, 6) 
y ≤ 3
y > –x + 5 
(6, 3)
(8, 1)
y ≤ 3
y > –x + 5
Graph the system.
(8, 1) and (6, 3) are solutions.
(–1, 4) and (2, 6) are not solutions.

9. Example 2B: Solving a System of Linear Inequalities by Graphing
Graph the system of linear inequalities. Give two ordered pairs that are solutions and two that are not solutions.
–3x + 2y ≥ 2
y < 4x + 3
–3x + 2y ≥ 2
Solve the first inequality for y.
2y ≥ 3x + 2

10. (–3, 1) and (–1, –4) are not solutions.
Check It Out! Example 2a
Graph the system of linear inequalities. Give two ordered pairs that are solutions and two that are not solutions. 
(3, 3)
(4, 4)
y ≤ x + 1
y > 2
(–3, 1) 
(–1, –4)
y ≤ x + 1
y > 2
Graph the system.
(3, 3) and (4, 4) are solutions.
(–3, 1) and (–1, –4) are not solutions.

11. Check It Out! Example 2b
Graph the system of linear inequalities. Give two ordered pairs that are solutions and two that are not solutions.
y > x – 7
3x + 6y ≤ 12
3x + 6y ≤ 12
Solve the second inequality for y.
6y ≤ –3x + 12
y ≤ x + 2

12. Example 3A: Graphing Systems with Parallel Boundary Lines
Graph the system of linear inequalities.
Describe the solutions.
y ≤ –2x – 4
y > –2x + 5
This system has no solutions.

13. Example 3B: Graphing Systems with Parallel Boundary Lines
Graph the system of linear inequalities. Describe the solutions.
y > 3x – 2
y < 3x + 6
The solutions are all points between the parallel lines but not on the dashed lines.

14. Example 3C: Graphing Systems with Parallel Boundary Lines
Graph the system of linear inequalities. Describe the solutions.
y ≥ 4x + 6
y ≥ 4x – 5
The solutions are the same as the solutions of y ≥ 4x + 6.

15. Check It Out! Example 3
Graph the system of linear inequalities. Describe the solutions.
y > x + 1
y ≤ x – 3
This system has no solutions.

16. Example 4: Application
In one week, Ed can mow at most 9 times and rake at most 7 times. He charges $20 for mowing and $10 for raking. He needs to make more than $125 in one week. Show and describe all the possible combinations of mowing and raking that Ed can do to meet his goal. List two possible combinations.
Earnings per Job ($)
Mowing 20
Raking 10

17. Check It Out! Example 4
At her party, Alice is serving pepper jack cheese and cheddar cheese. She wants to have at least 2 pounds of each. Alice wants to spend at most $20 on cheese. Show and describe all possible combinations of the two cheeses Alice could buy. List two possible combinations.
Price per Pound ($)
Pepper Jack 4
Cheddar 2