we use inequalities when there is a range of possible answers for a situation. i have to be there in...
TRANSCRIPT
We use inequalities when there is a range of possible answers for a situation.
“I have to be there in less than 5 minutes,”
“This team needs to score at least a goal to have a chance of winning,”
“To get into the city, I need at most $6.50 for train fare”
are all examples of situations where a limit is specified, but …
I < 5 minutes
T ≥ 1 goal
F ≤ $6.50
… a range of possibilities exist beyond that limit.
That’s what we are interested in when we study inequalities.
linear inequalities in two variableslinear inequalities in two variables3.73.7
Sara and Ali want to donate some money to a local food pantry. To raise funds, they are selling necklaces and earrings that they have made themselves. Necklaces cost $8 and earrings cost $5. What is the range of possible sales they could make in order to donate at least $100?
amount of money earned from selling
earrings
+amount of money
earned from selling necklaces
≥ $100
5y+8x ≥ $100
… The objective of today’s lesson is to graph such linear inequalities on the coordinate plane, and determine a solution set
Example 1
Graph each inequality
x + 2y ≤ 4
22
x
y2
1m
2
1m
2b 2b
2 2 2
Step1: Put the inequality in slope intercept form: y = mx + b and graph it.
Step1: Put the inequality in slope intercept form: y = mx + b and graph it.
2y ≤ -x + 4
b
Caution≥ or ≤ (solid)
< or >(dashed)IF
Step2 Select a testing point that’s not on the line, substitute it into the inequality. If the result is true, shade that region. Otherwise shade the opposite one.
Step2 Select a testing point that’s not on the line, substitute it into the inequality. If the result is true, shade that region. Otherwise shade the opposite one.
0 + 2(0 )≥ 4
0 ≥ 4 0 ≥ 4 True …
Test (0, 0) .. The easiest point
Example 2 3x ≥ 2)y – 1(
12
3
xy2
3m
2
3m
1b 1b
3x ≥ 2y – 2
b
- 2y ≥ -3x – 2
Caution≥ or ≤ (solid)
< or >(dashed)IF
-2 -2 -2
Test (0, 0)
0 ≤- 2 0 ≤- 2 true …
3)0( ≤ 2(0 – 1)
3x ≥ 2)y – 1(
Example 3
x + 1 < 0
x < – 1
Vertical line through x = – 1
Test (0, 0)
False … …so we shade the
region that does not contain point )0, 0(
x < – 1
0 < – 1 0 < – 1
Example 4
y – 2 < 0
y < 2
horizontal line through y = 2
Test (0, 0)
False … …so we shade the
region that does not contain point )0, 0(
y < 2
0 < 2 0 < 2
Example 5
x + y < 0
1
1m
1
1m
0b 0b
y < -x + 0
b
Test (-1, -1)
False …
-1-) + 1( < 0
-2 < 0 -2 < 0
Written Exercises .. page 138Written Exercises .. page 138
2( x – 1 < 0
y
x
4( y + 2 ≥ 0
y
x
6( x + y < 0
y
x
8( x + 2y ≥ 0
y
x
10( x – y < 1
y
x
Page 138
#s 12, 14, 16, 18
HomeworkHomeworkHomeworkHomework
12( x + 2y ≤ 2
y
x
14( 2x – 3y < 6
y
x
y
x
16( )4(2
1xy
18( 2)y – 1( < 3)x + 1(
y
x
Graph each system of inequalities
2
13
2
xy
xy 3
2m
3
2m 1b 1b
1
1m
1
1m 2b 2b
b
Test (0, 0)
True …
0 ≤ 0- 1
0 ≤- 1 0 ≤- 1
b
Test (0, 0)
True …
0 ≥ 0+ 2
0 ≥ 2 0 ≥ 2
Page 138
#s 20, 24, 28, 32
HomeworkHomeworkHomeworkHomework
Graph each system of inequalities
1
1m
1
1m 2b 2b
b
Test (0, 0)
True …
0 < 0 + 2
0 < 2 0 < 2
Test (0, 0)
False … 0 < 1 0 < 1
23(
01
2
y
xy
1 – y < 0
–y < – 1
y < 1 Horizontal line
y
x
20(
02
0
y
yx
y
x
24(
2
2
xy
xy
y
x
28(
022
022
yx
yx
y
x
32(
3
0
yx
xy
y