W5D1 Inequality Systems
Warm Up
1. 2x+4y < 12
2. y≥ 21 x-3
Graph the inequalities on the same axis
Solution:
5
4
3
2
1
−4 −3 −2 −1
−1
−2
−3
−4
−5
−6
−7
1
2
3
4
5
6
7
8
Figure 1: 1. 2x+4y < 12
2. y≥ 12 x-3
Lesson 12 - Systems of Inequalities
y ≤ 3x − 4
EX 1:
y >x+2
14
12
10
8
6
4
2
−2
−2
2
4
6
8
−4
Figure 2: y ≤ 3x − 4
EX 1B:
1
y ≥ |x + 4| − 2 and y < − x − 1
2
y >x+2
5
4
3
2
1
−10−9 −8 −7 −6 −5 −4 −3 −2 −1
−1
−2
−3
−4
−5
−6
−7
−8
1 2
y > 12 x + 5
EX 2:
y ≤ −x + 4
x ≥ −8
11
10
9
8
7
6
5
4
3
2
1
−10 −8
−6
−4
−2−1
−2
2
4
6
Figure 3: y < 12 x + 5
y < −x + 4
x ≥ −8
EX 3:
1. You can work at most 20 hours next week. You need to earn at least $92 to cover you weekly expenses. Your dog- walking
job pays $8 per hour and your job as a car wash attendant pays $6 per hour. Write a system of linear inequalities to model the
situation.
c+d ≤ 20
8d+6c ≥ 92
20
18
16
14
12
10
8
6
4
2
3
6
9
12
15
18
21
Figure 4: Ex3
EX 4:
The Community Service club is doing a fundraiser to sell candles. Small candles cost $3.00 and large candles
cost $5.00. They have at most 30 candles, and want to make at least $80. Write a system of linear inequalities that represent
the situation.
3s+5b ≥ 80
s+b ≤45
50
45
40
35
30
25
20
15
10
5
5
10 15 20 25 30 35 40
Figure 5: Ex 4
Linear Programming PPT
You arrived at school late because your car broke down, and therefore have only 45 minutes to complete a math test. The test
has 2 open-ended questions and 30 multiple-choice questions. Each correct open-ended question is worth 20 points, and each
multiple-choice question is worth 2 points. You know that it takes you 15 minutes to answer an open-ended question and only
1 minute to answer a multiple-choice question. Assume that you receive full credit on any question you answer.
0≤f ≤2
0 ≤ m ≤ 30
15f + 1m ≤ 45
20f + 2m = Score
Where f is free response question answered and m is multiple choice questions answered.
4
3
2
1
5
10 15 20 25 30 35 40
5
10 15 20 25 30 35 40
4
3
2
1
The vertices of the shaded region are (0, 2) (15, 2) (30, 1) and (30, 0)
( m, f)
2m + 20f
Score
(0, 2)
2(20
40
(15, 2)
2(15) + 20(2)
70
(30,1)
2(30)+20(1)
80
(30, 0)
2(30)
60
Answering 30 Multiple Choice and 1 Free Response will yield the highest score (80) on the test.
Complex Cookie Problem
The DHS Math teachers are selling cookies as a fundraiser. They want to figure out how to make the most money possible with
the given constraints. A dozen sugar cookies require 1 pound of dough and frosted cookies require 0.7 pounds of dough and 0.4
pounds of icing. You have 110 pounds of dough and 32 pounds of icing.
A dozen sugar cookies requires 0.1 hours of time to prepare and a dozen frosted cookies require 0.15 hours of preparation time.
You have 15 hours available to prepare the cookies.
The oven has room to cook 140 dozen cookies over the course of the 15 hours.
We know that all of the cookies will sell when we make them. The sugar cookies sell for $6.00 per dozen and the frosted cookies
sell for $7.00 per dozen. The sugar cookies cost $4.50 to make and the frosted cookies cost $5.00 to make (per dozen).
How many dozen of each kind of cookie should the math teachers make to maximize our profit?
Equations : s = sugar cookies
f = frosted cookies
Constraint
Equation
s intercept
f intercept
dough
1s + .7f ≤ 110
(110,0)
(0, 157.1)
Time
.1s + .15f ≤ 15
(150,0)
(0, 100)
Icing
.4f ≤ 32
none
(0, 80)
Oven
s + f ≤ 140
(140,0)
(0, 140)
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
160
150
140
130
120
110
100
90
80
70
60
50
40
30
20
10
20
40
60
80
100 120 140 160
Figure 6: Red = dough
Profit = 1.5 s+ 2 f
blue - time
20
40
green = frosting
60
80
100
black = oven
120
140
160
(s, f)
P = 1.5s+2f
Profit
(30, 80)
45+160
205
(0,80)
0+160
160
(75, 50)
112.5 + 100
212.5
(110, 0)
165 + 0
165
Easier Cookie Problem
You make different-sized packages of cookies that contain a combination of chocolate chip and peanut butter cookies. You place
at least three of each type of cookie in each of the combination packages. The largest package, the ”Maurer Power” contains up
to thirteen cookies.
It costs you 19 cents to make a chocolate chip cookie and 13 cents to make a peanut butter cookie. You sell them for 44 cents
and 39 cents, respectively. What is your profit equation going to look like?
c≥3
p≥3
c + p ≤ 13
14
13
12
11
10
9
8
7
6
5
4
3
2
1
14
13
12
11
10
9
8
7
6
5
4
3
2
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14
1 2 3 4 5 6 7 8 9 10 11 12 13 14
Profit Function is (44-19) c + (39-13) p = f(c, p)
f(c, p) = 25c+26 p
(c, p)
f(c, p)
(3,3)
153
(3, 10)
335
(10,3)
328
so 3 Chocolate Chip and 10 Peanut Better is the best deal
Extra Examples If needed
EX 4:
John is packing books into boxes. Each box can hold either 15 small books or 8 large books. He only has 100
s
b
boxes and but has to pack at least 350 books. s+b ≥ 350
15 + 8 ≤ 35
1,500
1,400
1,300
1,200
1,100
1,000
900
800
700
600
500
400
300
200
100
5
10
15
20
25
30
Figure 7: Ex 4
EX 8:
Maddie is buying plants and soil for her garden. The soil cost $4 per bag, and the plants cost $10 each. She
wants to buy at least 5 plants and can spend no more than $100. How many bags of soil and plants can she buy?
p≥5
4s+10p ≤ 100
15
10
5
5
10
15
20
25
30
Figure 8: Ex 5
EX 7:
During a family trip, you share the driving with your dad. At most, you are allowed to drive for three hours.
While driving, your maximum speed is 55 miles per hour. Write a system of inequalities describing the possible numbers of
hours t and distance d you may have to drive. Is it possible for you to have driven 160 miles?
t≤3
d ≤ 55 t
Go over the Quiz.
Unit 2 Learning Targets
Exit Pass
1. Robbie is buying wings and hot dogs for a party. One package of wings costs $8. Hot dogs cost $5 per pound. He must
spend less than $40. Robbie knows he will be buying at least 4 pound of hot dogs. Write a system of inequalities to model the
situation. Graph both inequalities and shade the intersection.
8w+5h ≤ 40
h >4
h leq − 85 w + 8
180
160
140
120
100
80
60
40
20
1
2
3
4
Figure 9: Ex 7
10
9
8
7
6
5
4
3
2
1
1
2
3
4
5
6
7
8
9
10
Figure 10: Exit Pass
Matching Activity
1. Noel must score at least 20 points to keep Coach Baker happy Shaheen and Noel together need to score at least 30 points to
win the game
Equation D Graph II
2. Juan wants a new phone. He has $30 and earns $5 an hour at his job (after taxes) Nathan wants the same phone. He has
$20 and earns $5 an hour at his job (after taxes)
Equation F Graph V
3. Kelsi paid $20 to get into the Santa Cruz Beach Boardwalk and then paid $4 per ride David bought a pass for $30 and only
pays $1 per ride
Equation A. Graph VI
4. Jeremiah started with $30 and spent $5 per day on snacks Bryan has $30 and spent $5 on iTunes every day after school
Equation E Graph III
5. Diana had 30 pencils and gave away 2 each day Marisol had 20 pencils and found 3 new ones every day
Equation B Graph I
6. Lauren has no more than 30 minutes to spend on homework for Math and English before the game Lauren also knows she
will spend at most 20 minutes on her math homework
Equation C Graph IV