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Algebra 1 Interactive Chalkboard
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Lesson 6-1 Solving Inequalities by Addition
and Subtraction
Lesson 6-2 Solving Inequalities by Multiplication
and Division
Lesson 6-3 Solving Multi-Step Inequalities
Lesson 6-4 Solving Compound Inequalities
Lesson 6-5 Solving Open Sentences Involving
Absolute Value
Lesson 6-6 Graphing Inequalities in Two Variables
Example 1 Solve by Adding
Example 2 Graph the Solution
Example 3 Solve by Subtracting
Example 4 Variables on Both Sides
Example 5 Write and Solve an Inequality
Example 6 Write an Inequality to Solve a Problem
Solve
Then check your solution.
Original inequality
Add 12 to each side.
This means all numbers
greater than 77.
Check Substitute 77, a number less than 77, and a
number greater than 77.
Answer: The solution is the set
{all numbers greater than 77}.
Solve
Then check your solution.
Answer:
or {all numbers less than 14}
Solve
Then graph it on a number line.
Original inequality
Add 9 to each side.
Simplify.
Answer: Since
is the same as y  21,
the solution set is
The heavy arrow pointing
to the left shows that the
inequality includes all the
numbers less than 21.
The dot at 21
shows that 21
is included in
the inequality.
Solve
Answer:
Then graph it on a number line.
Solve
Then graph the solution.
Original inequality
Subtract 23 from each side.
Simplify.
Answer: The solution set is
Solve
Answer:
Then graph the solution.
Then graph the solution.
Original inequality
Subtract 12n from each side.
Simplify.
Answer: Since
is the same as
solution set is
the
Then graph the solution.
Answer:
Write an inequality for the sentence below. Then
solve the inequality.
Seven times a number is greater than 6 times that number
minus two.
Seven times is greater
six times
a number
than
that number minus
two.
7n
>
6n
–
2
Original inequality
Subtract 6n from each side.
Simplify.
Answer: The solution set is
Write an inequality for the sentence below. Then
solve the inequality.
Three times a number is less than two times that number
plus 5.
Answer:
Entertainment Alicia wants to buy season passes to
two theme parks. If one season pass costs $54.99,
and Alicia has $100 to spend on passes, the second
season pass must cost no more than what amount?
Words
The total cost of the two passes must be
less than or equal to $100.
Variable
Let
Inequality
the cost of the second pass.
is less than
The total cost
or equal to
$100.
100
Solve the inequality.
Original inequality
Subtract 54.99 from
each side.
Simplify.
Answer: The second pass must cost no more than $45.01.
Michael scored 30 points in the four rounds of the
free throw contest. Randy scored 11 points in the
first round, 6 points in the second round, and 8 in
the third round. How many points must he score in
the final round to surpass Michael’s score?
Answer: 6 points
Example 1 Multiply by a Positive Number
Example 2 Multiply by a Negative Number
Example 3 Write and Solve an Inequality
Example 4 Divide by a Positive Number
Example 5 Divide by a Negative Number
Example 6 The Word “not”
Then check your solution.
Original inequality
Multiply each side by 3. Since we
multiplied by a positive number, the
inequality symbol stays the same.
Simplify.
Check To check this solution, substitute 36, a number
less than 36, and a number greater than 36 into
the inequality.
Answer: The solution set is
Then check your solution.
Answer:
Original inequality
Multiply each side by
change
Simplify.
Answer: The solution set is
and
Answer:
Write an inequality for the sentence below. Then
solve the inequality.
Four-fifths of a number is at most twenty.
Four-fifths
of
a number

r
is at most
twenty.
20
Original inequality
Multiple each side by
and do not
change the inequality’s direction.
Simplify.
Answer: The solution set is
.
Write an inequality for the sentence below. Then
solve the inequality.
Two-thirds of a number is less than 12.
Answer:
Original inequality
Divide each side by 12 and do not
change the direction of the inequality
sign.
Simplify.
Check
Answer: The solution set is
Answer:
using two methods.
Method 1 Divide.
Original inequality
Divide each side by –8 and
change < to >.
Simplify.
Method 2 Multiply by the multiplicative inverse.
Original inequality
Multiply each side by
and change < to >.
Simplify.
Answer: The solution set is
using two methods.
Answer:
Multiple-Choice Test Item
Which inequality does not have the solution
A
B
C
D
Read the Test Item
You want to find the inequality that does not have the
solution set
Solve the Test Item
Consider each possible choice.
A.
B.
C.
D.
Answer: B
Multiple-Choice Test Item
Which inequality does not have the solution
A
Answer: C
B
C
D
?
Example 1 Solve a Real-World Problem
Example 2 Inequality Involving a Negative Coefficient
Example 3 Write and Solve an Inequality
Example 4 Distributive Property
Example 5 Empty Set
Science The inequality F > 212 represents the
temperatures in degrees Fahrenheit for which water is
a gas (steam). Similarly, the inequality
represents the temperatures in degrees Celsius for
which water is a gas. Find the temperature in degrees
Celsius for which water is a gas.
Original inequality
Subtract 32 from each side.
Simplify.
Multiply each side by
Simplify.
Answer: Water will be a gas for all temperatures
greater than 100°C.
Science The boiling point of helium is –452°F. Solve
the inequality
to find the temperatures
in degrees Celsius for which helium is a gas.
Answer: Helium will be a gas for all temperatures
greater than –268.9°C.
Then check your solution.
Original inequality
Subtract 13 from each side.
Simplify.
Divide each side by –11 and
change
Simplify.
Check To check the solution, substitute –6, a number
less than –6, and a number greater than –6.
57
Answer: The solution set is
Then check your solution.
Answer:
Write an inequality for the sentence below. Then solve
the inequality.
Four times a number plus twelve is less than the number
minus three.
Four times
a number
plus
twelve
is less
than
4n
+
12
<
the number
minus three.
Original inequality
Subtract n from each side.
Simplify.
Subtract 12 from each side.
Simplify.
Divide each side by 3.
Simplify.
Answer: The solution set is
Write an inequality for the sentence below. Then solve
the inequality.
6 times a number is greater than 4 times the number
minus 2.
Answer:
Original inequality
Distributive Property
Combine like terms.
Add c to each side.
Simplify.
Subtract 6 from each side.
Simplify.
Divide each side by 4.
Simplify.
Answer: Since
is the same as
the solution set is
Answer:
Original inequality
Distributive Property
Combine like terms.
Subtract 4s from each side.
This statement is false.
Answer: Since the inequality results in a false
statement, the solution set is the empty set Ø.
Answer: Ø
Example 1 Graph an Intersection
Example 2 Solve and Graph an Intersection
Example 3 Write and Graph a Compound Inequality
Example 4 Solve and Graph a Union
Graph the solution set of
Graph
Graph
Find the intersection.
Answer: The solution set is
Note that the
graph of
includes the point 5. The graph
of
does not include 12.
Graph the solution set of
and
Then graph the solution set.
First express
inequality.
using and. Then solve each
and
The solution set is the intersection of the two graphs.
Graph
Graph
Find the intersection.
Answer: The solution set is
Then graph the solution set.
Answer:
Travel A ski resort has several types of hotel rooms
and several types of cabins. The hotel rooms cost at
most $89 per night and the cabins cost at least $109
per night. Write and graph a compound inequality
that describes the amount that a guest would pay
per night at the resort.
Words
The hotel rooms cost at most $89 per night and
the cabins cost at least $109 per night.
Variables Let c be the cost of staying at the resort
per night.
Inequality Cost per is at
the is at
night
c
most $89 or cost least $109.
89
or
c
109
Now graph the solution set.
Graph
Graph
Find the union.
Answer:
Ticket Sales A professional hockey arena has seats
available in the Lower Bowl level that cost at most
$65 per seat. The arena also has seats available at
the Club Level and above that cost at least $80 per
seat. Write and graph a compound inequality that
describes the amount a spectator would pay for a
seat at the hockey game.
Answer:
where c is the cost per seat
Then graph the solution set.
or
Graph
Graph
Answer:
Notice that the graph of
contains every point in
the graph of
So, the union is the graph of
The solution set is
Then graph the solution set.
Answer:
Example 1 Solve an Absolute Value Equation
Example 2 Write an Absolute Value Equation
Example 3 Solve an Absolute Value Inequality (<)
Example 4 Solve an Absolute Value Inequality (>)
Method 1 Graphing
means that the distance between b and –6
is 5 units. To find b on the number line, start at –6 and
move 5 units in either direction.
The distance from –6 to –11 is 5 units.
The distance from –6 to –1 is 5 units.
Answer: The solution set is
Method 2 Compound Sentence
Write
as
or
Case 1
Case 2
Original inequality
Subtract 6 from
each side.
Simplify.
Answer: The solution set is
Answer: {12, –2}
Write an equation involving the absolute value for
the graph.
Find the point that is the same distance from –4 as the
distance from 6. The midpoint between –4 and 6 is 1.
The distance from 1 to –4 is 5 units.
The distance from 1 to 6 is 5 units.
So, an equation is
.
Answer:
Check Substitute –4 and 6 into
Write an equation involving the absolute value for
the graph.
Answer:
Then graph the solution set.
Write
as
and
Case 2
Case 1
Original inequality
Add 3 to each side.
Simplify.
Answer: The solution set is
Then graph the solution set.
Answer:
Then graph the solution set.
Write
as
or
Case 2
Case 1
Original inequality
Add 3 to each side.
Simplify.
Divide each side by 3.
Simplify.
Answer: The solution set is
Then graph the solution set.
Answer:
Example 1 Ordered Pairs that Satisfy an Inequality
Example 2 Graph an Inequality
Example 3 Write and Solve an Inequality
From the set {(3, 3), (0, 2), (2, 4), (1, 0)}, which ordered
pairs are part of the solution set for
Use a table to substitute the x and y values of each
ordered pair into the inequality.
x
y
True or False
3
3
true
0
2
false
2
4
true
1
0
false
Answer: The ordered pairs {(3, 3), (2, 4)} are part of the
solution set of
. In the graph, notice
the location of the two ordered pairs that are
solutions for
in relation to the line.
From the set {(0, 2), (1, 3), (4, 17), (2, 1)}, which
ordered pairs are part of the solution set for
Answer: {(1, 3), (2, 1)}
Step 1 Solve for y in terms of x.
Original inequality
Add 4x to each side.
Simplify.
Divide each side by 2.
Simplify.
Step 2 Graph
Since
does not include
values when
the
boundary is not included in the
solution set. The boundary
should be drawn as a dashed line.
Step 3 Select a point in one of
the half-planes and test it.
Let’s use (0, 0).
Original inequality
false
y = 2x + 3
Answer: Since the statement is
false, the half-plane containing the
origin is not part of the solution.
Shade the other half-plane.
y = 2x + 3
Answer: Since the statement is
false, the half-plane containing the
origin is not part of the solution.
Shade the other half-plane.
Check Test the point in the
other half-plane, for example,
(–3, 1).
Original inequality
Since the statement is true, the half-plane
containing (–3, 1) should be shaded. The
graph of the solution is correct.
y = 2x + 3
Answer:
Journalism Lee Cooper writes and edits short
articles for a local newspaper. It generally takes her
an hour to write an article and about a half-hour to
edit an article. If Lee works up to 8 hours a day, how
many articles can she write and edit in one day?
Step 1 Let x equal the number of articles Lee can write.
Let y equal the number of articles that Lee can edit.
Write an open sentence representing the situation.
Number of
articles
she can write plus
x
+
hour times
number of
articles
she can edit is up to 8 hours.
y
8
Step 2 Solve for y in terms of x.
Original inequality
Subtract x from each side.
Simplify.
Multiply each side by 2.
Simplify.
Step 3 Since the open sentence includes the equation,
graph
as a solid line. Test a point in
one of the half-planes, for example, (0, 0). Shade
the half-plane containing (0, 0) since
is true.
Answer:
Step 4 Examine the situation.
 Lee cannot work a negative number of hours.
Therefore, the domain and range contain only
nonnegative numbers.
 Lee only wants to count articles that are
completely written or completely edited. Thus,
only points in the half-plane whose x- and ycoordinates are whole numbers are
possible solutions.
 One solution is (2, 3). This represents 2 written
articles and 3 edited articles.
Food You offer to go to the local deli and pick up
sandwiches for lunch. You have $30 to spend. Chicken
sandwiches cost $3.00 each and tuna sandwiches are
$1.50 each. How many sandwiches can you purchase
for $30?
Answer:
The open sentence that represents this situation is
where x is the number of chicken
sandwiches, and y is the number of tuna sandwiches.
One solution is (4, 10). This means that you could
purchase 4 chicken sandwiches and 10 tuna sandwiches.
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