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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
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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
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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
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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(
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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
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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
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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
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Written Exercises .. page 138Written Exercises .. page 138
2( x – 1 < 0
y
x
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4( y + 2 ≥ 0
y
x
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6( x + y < 0
y
x
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8( x + 2y ≥ 0
y
x
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10( x – y < 1
y
x
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Page 138
#s 12, 14, 16, 18
HomeworkHomeworkHomeworkHomework
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12( x + 2y ≤ 2
y
x
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14( 2x – 3y < 6
y
x
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y
x
16( )4(2
1xy
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18( 2)y – 1( < 3)x + 1(
y
x
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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
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Page 138
#s 20, 24, 28, 32
HomeworkHomeworkHomeworkHomework
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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
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y
x
20(
02
0
y
yx
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y
x
24(
2
2
xy
xy
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y
x
28(
022
022
yx
yx
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y
x
32(
3
0
yx
xy
y