Warm Up

1
Warm Up
Problem of the Day
Lesson Presentation
2
Warm Up Solve. 1. 21z 12 27z 2. 12n 18
6n 3. 12y 56 8y 4. 36k 9 18k
z 2
n 3
y 14
1 2
k
3
Problem of the Day The dimensions of one
rectangle are twice as large as the dimensions of
another rectangle. The difference in area is 42
cm2. What is the area of each rectangle?
56 cm2 and 14 cm2
4
Learn to read and write inequalities and graph
them on a number line.
5
Insert Lesson Title Here
Vocabulary
inequality algebraic inequality solution
set compound inequality
6
An inequality states that two quantities either
are not equal or may not be equal. An inequality
uses one of the following symbols
Symbol Meaning Word Phrases
lt
gt


is less than
Fewer than, below
is greater than
More than, above
is less than or equal to
At most, no more than
is greater than or equal to
At least, no less than
7
Additional Example 1 Writing Inequalities
Write an inequality for each situation.
A. There are at least 15 people in the waiting
room.
At least means greater than or equal to.
number of people 15
B. The tram attendant will allow no more
than 60 people on the tram.
No more than means less than or equal to.
number of people 60
8
Check It Out Example 1
Write an inequality for each situation.
A. There are at most 10 gallons of gas in the
tank.
At most means less than or equal to.
gallons of gas 10
B. There is at least 10 yards of fabric left.
At least means greater than or equal to.
yards of fabric 10
9
An inequality that contains a variable is an
algebraic inequality. A value of the variable
that makes the inequality true is a solution of
the inequality.
An inequality may have more than one solution.
Together, all of the solutions are called the
solution set.
You can graph the solutions of an inequality on a
number line. If the variable is greater than or
less than a number, then that number is
indicated with an open circle.
10
This open circle shows that 5 is not a solution.
a gt 5
If the variable is greater than or equal to or
less than or equal to a number, that number is
indicated with a closed circle.
This closed circle shows that 3 is a solution.
b 3
11
Additional Example 2 Graphing Simple Inequalities
Graph each inequality.
A. n lt 3
3 is not a solution, so draw an open circle at
3. Shade the line to the left of 3.
3 2 1 0 1 2 3
B. a 4
4 is a solution, so draw a closed circle at 4.
Shade the line to the right of 4.
6 4 2 0 2 4 6
12
Check It Out Example 2
Graph each inequality.
A. p 2
2 is a solution, so draw a closed circle at 2.
Shade the line to the left of 2.
3 2 1 0 1 2 3
B. e gt 2
2 is not a solution, so draw an open circle at
2. Shade the line to the right of 2.
3 2 1 0 1 2 3
13
A compound inequality is the result of combining
two inequalities. The words and and or are used
to describe how the two parts are related.
x gt 3 or x lt 1
2 lt y and y lt 4
x is either greater than 3 or less than1.
y is both greater than 2 and less than 4. y
is between 2 and 4.
14
Additional Example 3A Graphing Compound
Inequalities
Graph each compound inequality.
m 2 or m gt 1
First graph each inequality separately.
m 2
m gt 1

º
Then combine the graphs.
The solutions of m 2 or m gt 1 are the combined
solutions of m 2 or m gt 1.
15
Additional Example 3B Graphing Compound
Inequalities
Graph each compound inequality
3 lt b 0
3 lt b 0 can be written as the inequalities 3
lt b and b 0. Graph each inequality separately.
3 lt b
b 0

º
Then combine the graphs. Remember that
3 lt b 0 means that b is between 3 and 0,
and includes 0.
16
(No Transcript)
17
Check It Out Example 3A
Graph each compound inequality.
w lt 2 or w 4
First graph each inequality separately.
w lt 2
W 4
Then combine the graphs.
The solutions of w lt 2 or w 4 are the combined
solutions of w lt 2 or w 4.
18
Check It Out Example 3B
Graph each compound inequality
5 gt g 3
5 gt g 3 can be written as the inequalities 5 gt
g and g 3. Graph each inequality separately.
5 gt g
g 3

º
Then combine the graphs. Remember that
5 gt g 3 means that g is between 5 and 3,
and includes 3.
19
Insert Lesson Title Here
Lesson Quiz Part I
Write an inequality for each situation. 1. No
more than 220 people are in the theater. 2.
There are at least a dozen eggs left. 3. Fewer
than 14 people attended the meeting.
people in the theater 220
number of eggs 12
people attending the meeting lt 14
20
Insert Lesson Title Here
Lesson Quiz Part II
Graph the inequalities. 4. x gt 1
5. x 4 or x lt 1
21
Warm Up
Problem of the Day
Lesson Presentation
22
Warm Up Write the inequality for each
situation. 1. There are at least 28 days in a
month. 2. The temperature is above 72. 3. At
most 9 passengers can ride in the van.
days in a month 28
temperature gt 72
passengers 9
23
Problem of the Day Daryl gave the clerk less
than 20 for a CD and received change of at least
5. He ended up with the CD and less money than
he started with. Write a compound inequality to
show what C, the cost in dollars of the CD, could
have been.
0 lt C lt 15
24
Learn to solve one-step inequalities by adding or
subtracting.
25
When you add or subtract the same number on both
sides of an inequality, the resulting statement
will still be true.
2 lt 5
7 7
5 lt 12
You can find solution sets of inequalities the
same way you find solutions of equations, by
isolating the variable.
26
Additional Example 1A Solving Inequalities by
Adding
Solve. Then graph the solution set on a number
line.
n 7 15
n 7 15
7 7
Add 7 to both sides.
n 22
Draw a closed circle at 22 then shade the line
to the left of 22.
27
Additional Example 1B Solving Inequalities by
Adding
Solve. Then graph the solution set on a number
line.
a 10 3
a 10 3
10 10
Add 10 to both sides.
a 7
Draw a closed circle at 7. Then shade the line
to the right.
28
Check It Out Example 1A
Solve. Then graph the solution set on a number
line.
d 12 18
d 12 18
12 12
Add 12 to both sides.
d 6
Draw a closed circle at 6 then shade the line
to the left of 6.
29
Check It Out Example 1B
Solve. Then graph the solution set on a number
line.
b 14 8
b 14 8
14 14
Add 14 to both sides.
b 6
Draw a closed circle at 6. Then shade the line
to the right.
30
You can check the solution to an inequality is
true by choosing any number in the solution set
and substituting it into the original inequality.
31
Additional Example 2A Solving Inequalities by
Subtracting
Solve. Check each answer.
d 11 gt 6
d 11 gt 6
11 11
Subtract 11 from both sides.
d gt 5
Check
d 11 gt 6
0 is greater than 5. Substitute 0 for d.
?
0 11 gt 6
?
11 gt 6
32
Additional Example 2B Solving Inequalities by
Subtracting
Solve. Check your answer.
b 12 19
b 12 19
12 12
Subtract 12 from both sides.
b 7

Check
b 12 19
6 12 19
6 is less than 7. Substitute 6 for b.
18 19
33
Check It Out Example 2A
Solve. Check each answer.
c 15 gt 9
c 15 gt 9
15 15
Subtract 15 from both sides.
c gt 6
Check
c 15 gt 9
0 is greater than 6. Substitute 0 for c.
?
0 15 gt 9
?
15 gt 9
34
Check It Out Example 2B
Solve. Check your answer.
a 15 20
a 15 20
15 15
Subtract 15 from both sides.
a 5

Check
a 15 20
4 15 20
4 is less than 5. Substitute 4 for a.
19 20
35
(No Transcript)
36
Additional Example 3 Money Application
Edgars August profit of 137 was at least 20
higher than his July profit. What was Julys
profit?
Let p represent the profit increase from July to
August.
August profit was at least 20 higher than Julys
profit.
137 20
p
137 20 p
-20 -20
Subtract 20 from both sides.
117 p
p 117
Rewrite the inequality.
Julys profit was at most 117.
37
Check It Out Example 3
Rylans March profit of 172 was at least 12
less than his February profit. What was
Februarys profit?
Let p represent the profit decrease from February
to march.
March profit was at least 12 less than
Februarys profit.
172 -12
p
172 -12 p
12 12
Add 12 to both sides.
184 p
p 184
Rewrite the inequality.
Februarys profit was at most 184.
38
Lesson Quiz Part I
Solve. Then graph each solution set on a number
line. 1. x 4 gt 17 2. z 27 19 Solve.
Check each answer 3. p 18 6
p 24
k gt 18
4. k 47 gt 65
39
Lesson Quiz Part II
Solve. Check each answer. 5. There are at least
17 more bus riders than walkers in a class. If
there are 7 walkers, how many bus riders are
there?
bus riders 24