-
Section 4.6 : Systems of Linear Inequalities
Learning Targets: A.REI.12, A.CED.3
Important Terms and Definitions
System of Linear Inequalities: made up of two or more linear
inequalities Solution of a System of Linear Inequalities: makes
each inequality in the system true
Let’s graph more than one inequality on the same coordinate
plane.
𝑦 > −2
𝑥 ≤ 3
What would the solution(s) be?
Identifying Solutions of a Linear Inequality
Example: Is (2,12) a solution of {𝑦 > 2𝑥 + 4𝑦 < 3𝑥 + 7
?
𝑦 > 2𝑥 + 4 𝑦 < 3𝑥 + 7
Is 12 > 2(2) + 4 ? Is 12 < 3(2) + 7 ?
12 > 4 + 4 12 < 6 + 7
12 > 8 12 < 13
TRUE TRUE
(2,12) is a solution of {𝑦 > 2𝑥 + 4𝑦 < 3𝑥 + 7
-
(ex 1) Is (1,−3) a solution to the system {𝑥 + 𝑦 < −1𝑥 − 𝑦
> 1
(ex 2) Is (−2,3) a solution to the system {2𝑥 + 𝑦 ≥ −1𝑥 + 3𝑦
> 10
Graphing a System of Inequalities
1. Graph each inequality, one at a time, just like you have done
before. (Remember the rules with dashed and solid lines, and
shading above and below) Hint: You may want to shade in different
colors or styles to make it easier! ☺
2. Look to see where the shading overlaps. This is the
solution.
-
(ex 3) Solve the system by graphing. {𝑦 < 2𝑥 + 3𝑦 > 𝑥 −
1
(ex 4) Solve the system by graphing. { 𝑦 ≥ −𝑥 + 22𝑥 + 4𝑦 <
4
(ex 5) Solve the system by graphing. {𝑦 ≤ 0.75𝑥 − 2𝑦 > 0.75𝑥
− 3
(ex 6) Solve the system by graphing. {3𝑥 + 2𝑦 ≤ 6𝑥 < 2
-
(ex 7) Write a system of inequalities for the shaded region
below. Is the point where the lines intersect a solution to the
system? Explain why or why not.
(ex 8) Jenna spends at most 150 minutes a night on math and
science homework. She spends at least 60 minutes on math. Write and
graph a system of inequalities to model how she allots her time for
these two subjects.
(ex 9) Andy plans to invest money in two different accounts; a
standard savings account and a riskier money market fund. The total
of the two investments can be no more than $1000. At least $300 is
to be put into the money market fund. Write and graph a system of
inequalities to model how he allots his money in the two
accounts.
-
Homework – Section 4.6 : Systems of Linear Inequalities
Is the given ordered pair a solution of the system of
inequalities?
1. (1, 19) 2. (−2, 40) 3. (7,−4) 𝑦 ≤ 7𝑥 − 13 𝑦 > −13𝑥 + 29 3𝑥
+ 4𝑦 ≤ 15 𝑦 > 3𝑥 + 6 𝑦 ≤ 9𝑥 + 11 𝑦 > 1
2𝑥 − 12
Solve each system of inequalities by graphing.
4. 𝑦 < 2𝑥 − 3 8. 2𝑥 − 3𝑦 < 6 𝑦 > −𝑥 + 2 3𝑥 + 4𝑦 <
12
5. 𝑥 + 𝑦 > 5 9. 𝑥 − 3𝑦 ≥ 3 𝑥 − 𝑦 ≤ 3 𝑥 + 2𝑦 < 4
6. 3𝑥 − 2𝑦 > 6 10. 3𝑥 − 5𝑦 ≤ 15 𝑥 + 𝑦 ≤ 4 3𝑥 + 2𝑦 < 12
7. 3𝑥 + 2𝑦 < 2 11. 𝑥 − 𝑦 ≥ 4 −𝑥 − 2𝑦 > 4 2𝑦 ≥ 𝑥 − 4
Write a system of inequalities for each graph.
12. 13.
14. Suppose you want to fence a rectangular area for your dog.
You will use the house as one of your four sides. Since the house
is 40 ft wide, the length needs to be no more than 40 ft. You plan
to use at least 150 ft of fencing. Write and graph a system of
equations to represent the possible dimensions for the rectangular
area.
15. Suppose you get a $40 iTunes gift certificate. You plan to
buy some songs and at least one episode of your favorite television
show. On average, a song will cost $0.99, and a television show
will cost $2.99. Write a system of inequalities for x songs and y
television shows to represent the situation. Graph the system to
show all possible solutions. What purchase does the ordered pair
(3, 5) represent? Is it a solution to your system?
