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Mathematics: Applications and Concepts, Course 3 Interactive Chalkboard
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Lesson 10-1
Simplifying Algebraic Expressions
Lesson 10-2
Solving Two-Step Equations
Lesson 10-3
Writing Two-Step Equations
Lesson 10-4
Solving Equations with Variables on
Each Side
Lesson 10-5
Inequalities
Lesson 10-6
Solving Inequalities by Adding or
Subtracting
Lesson 10-7
Solving Inequalities by Multiplying or
Dividing
Example 1 Write Equivalent Expressions
Example 2 Write Equivalent Expressions
Example 3 Write Expressions with Subtraction
Example 4 Write Expressions with Subtraction
Example 5 Identify Parts of an Expression
Example 6 Simplify Algebraic Expressions
Example 7 Simplify Algebraic Expressions
Example 8 Simplify Algebraic Expressions
Example 9 Translate Phrases into Expressions
Use the Distributive Property to rewrite
Simplify.
Answer:
.
Use the Distributive Property to rewrite
Answer:
Use the Distributive Property to rewrite
Simplify.
Answer:
Use the Distributive Property to rewrite
Answer:
Use the Distributive Property to rewrite
Rewrite
Distributive Property
Simplify.
Definition of subtraction
Answer:
Use the Distributive Property to rewrite
Answer:
Use the Distributive Property to rewrite
Rewrite
Distributive Property
Simplify.
Answer:
Use the Distributive Property to rewrite
Answer:
Identify the terms, like terms, coefficients, and
constants in
Definition of
subtraction
Identity Property;
Answer:
The like terms are
The coefficients are
The constant is –5.
Identify the terms, like terms, coefficients, and
constants in
Answer: The terms are
The like terms are
The coefficients are
The constant is –2.
Simplify 6n – n.
6n and n are like terms.
Identity Property;
Distributive Property
Simplify.
Answer:
Simplify 7n  n.
Answer:
Simplify 5s  3 – 12s.
are like terms.
Commutative Property
Distributive Property
Answer:
Simplify 6s  2 – 10s.
Answer:
Simplify 8z  z – 5 – 9z  2.
are like terms. –5 and 2 are also like terms.
Definition of
subtraction
Commutative
Property
Distributive
Property
Simplify.
Answer: –3
Simplify 6z  z – 2 – 8z  2.
Answer: –z
THEATER Tickets for the school play cost $5 for
adults and $3 for children. A family has the same
number of adults as children. Write an expression in
simplest form that represents the total amount of
money spent on tickets.
If x represents the number of adult tickets, then x also
represents the number of children tickets. To find the total
amount spent, multiply the cost of each ticket by the
number of tickets purchased. Then add the expressions.
Distributive Property
Simplify.
Answer: The expression $8x represents the total
amount of money spent on tickets, where
x is the number of adults or children.
MUSEUM Tickets for the museum cost $10 for
adults and $7.50 for children. A group of people have
the same number of adults as children. Write an
expression in simplest form that represents the total
amount of money spent on tickets to the museum.
Answer: $17.50x
Example 1 Solve a Two-Step Equation
Example 2 Solve Two-Step Equations
Example 3 Solve Two-Step Equations
Example 4 Equations with Negative Coefficients
Solve
Method 1 Use a model.
Remove 1-tile from the mat.
Separate the remaining tiles into 5 equal groups.
There are 5 tiles in each group.
Method 2 Use symbols.
Use the Subtraction Property of Equality.
Write the equation.
Subtract 1 from each side.
Use the Division Property of Equality.
Divide each side by 5.
Simplify.
Answer: The solution is 5.
Solve
Answer: 6
Solve
Check your solution.
Method 1 Vertical Method
Write the equation.
Add 8 to each side.
Simplify.
Divide each side by 2.
Simplify.
Method 2 Horizontal Method
Write the equation.
Add 8 to each side.
Simplify.
Divide each side by 2.
Simplify.
Check
Write the equation.
Replace n with 21.
The sentence is true.
Answer: The solution is 21.
Solve
Answer: 9
Check your solution.
Solve
Write the equation.
Subtract 2 from each side.
Simplify.
Multiply each side by 3.
Simplify.
Answer: The solution is –18.
Solve
Answer: –26
Solve
Write the equation.
Definition of subtraction
Subtract 8 from each side.
Simplify.
Divide each side by –3.
Simplify.
Answer: The solution is –2.
Solve
Answer: –3
Solve
Check your solution.
Write the equation.
Identity Property;
Combine like terms;
Add 2 to each side.
Simplify.
Divide each side by 2.
Simplify.
Check
Write the equation.
Replace k with 8.
Multiply.
The statement is true.
Answer: The solution is 8.
Solve
Answer: 5
Example 1 Translate Sentences into Equations
Example 2 Translate Sentences into Equations
Example 3 Translate Sentences into Equations
Example 4 Translate and Solve an Equation
Example 5 Write and Solve a Two-Stop Equation
Translate three more than half a number is 15 into an
equation.
Answer:
Translate five more than one-third a number is 7 into
an equation.
Answer:
Translate nineteen is two more than five times a
number into an equation.
Answer:
Translate fifteen is three more than six times a number
into an equation.
Answer:
Translate eight less than twice a number is –35 into
an equation.
Answer:
Translate six less than three times a number is –22
into an equation.
Answer:
Two more than
Words
Variable
Equation
of a number is 6. Find the number.
Two more than
of a number is 6.
Write the equation.
Subtract 2 from each
side.
Simplify.
Mentally multiply each side by 3.
Answer: The number is 12.
Three more than six times a number is 15. Find the
number.
Answer: 2
TRANSPORTATION A taxi ride costs $3.50 plus $2
for each mile traveled. If Jan pays $11.50 for the ride,
how many miles did she travel?
Her cost starts at $3.50 and adds $2 until it reaches
$11.50. Organize the data for the first few miles into a table
and look for a pattern.
Miles
0
1
2
3
Cost
Write an equation to represent the situation. Let m
represent the number of miles.
flat rate
plus
m miles at
$2 per
mile
3.50
+
2m
equals
$11.50
11.50
Write the equation.
Subtract 3.50 from
each side.
Simplify.
Divide each side by 2.
Simplify.
Answer: Jan traveled 4 miles.
TRANSPORTATION A rental car costs $100 plus
$0.25 for each mile traveled. If Kaya pays $162.50 for
the car, how many miles did she travel?
Answer: 250 miles
Example 1 Equations with Variables on Each Side
Example 2 Equations with Variables on Each Side
Example 3 Use an Equation to Solve a Problem
Solve
Check your solution.
Write the equation.
Subtract 7x from each side.
Simplify by combining like terms.
Mentally divide each side by 2.
To check your solution, replace x with 2 in the original
equation.
Check
Write the equation.
Replace x with 2.
The sentence is true.
Answer: The solution is 2.
Solve
Answer: ─3
Check your solution.
Solve
Write the equation.
Subtract 8x from each side.
Simplify.
Add 2 to each side.
Simplify.
Mentally divide each side
by –5.
Answer: The solution is –3.
Solve
Answer: –10
GRID-IN TEST ITEM Find the value of x so that the
polygons have the same perimeter.
Read the Test Item
You need to find the value of x that will make the
perimeter of the triangle equal to the perimeter of the
rectangle.
Solve the Test Item
Write expressions for the perimeter of each figure.
Then set the two expressions equal to each other and
solve for x.
Triangle
Rectangle
Answer:
GRID-IN TEST ITEM Find the value of x so that the
polygons have the same perimeter.
Answer:
Example 1 Write Inequalities with < or >
Example 2 Write Inequalities with < or >
Example 3 Write Inequalities with  or 
Example 4 Write Inequalities with  or 
Example 5 Determine the Truth of an Inequality
Example 6 Determine the Truth of an Inequality
Example 7 Graph an Inequality
Example 8 Graph an Inequality
SPORTS Members of the little league team must be
under 14 years old. Write an inequality for the
sentence.
Answer:
SPORTS Members of the peewee football team must
be under 10 years old. Write an inequality for the
sentence.
Answer:
CONSTRUCTION The ladder must be over 30 feet tall
to reach the top of the building. Write an inequality
for the sentence.
Answer:
CONSTRUCTION The new building must be over
300 feet tall. Write an inequality for the sentence.
Answer:
POLITICS The president of the United States must be
at least 35. Write an inequality for the sentence.
Answer:
VOTING To vote, you must be at least 18 years old.
Write an inequality for the sentence.
Answer:
CAPACITY A theater can hold a maximum of 300
people. Write an inequality for the sentence.
Answer:
CAPACITY A football stadium can hold a maximum
of 10,000 people. Write an inequality for the sentence.
Answer:
For the given value, state whether the inequality is
true or false.
Write the inequality.
Replace x with 0.
Simplify.
Answer: Since –4 is less than 6,
For the given value, state whether the inequality is
true or false.
Answer: false
For the given value, state whether the inequality is
true or false.
Write the inequality.
Replace x with 1.
Simplify.
Answer: Since 3 is not greater than or equal to 4,
the sentence is false.
For the given value, state whether the inequality is
true or false.
Answer: true
Graph n
–1 on a number line.
Place a closed circle at –1. Then draw a line and an
arrow to the left.
Answer:
The closed circle
means the number –1
is included in the
graph.
Graph n
Answer:
–3 on a number line.
Graph n
–1 on a number line.
Place an open circle at –1. Then draw a line and an arrow
to the right.
Answer:
The open circle means
–1 is not included in
the graph.
Graph n
Answer:
–3 on a number line.
Example 1 Solve an Inequality Using Addition
Example 2 Solve an Inequality Using Subtraction
Example 3 Graph the Solutions of an Inequality
Example 4 Use an Inequality to Solve a Problem
Solve
Check your solution.
Write the inequality.
Add 4 to each side.
Simplify.
Check
Write the inequality.
Replace n with a number greater
than 10, such as 11.
The statement is true.
Answer: Any number greater than 10 will make the
statement true, so the solution is
Solve
Answer:
Check your solution.
Solve
Check your solution.
Write the inequality.
Subtract 8 from each side.
Simplify.
Check Replace x in the original inequality with –15 and
then with a number less than –15.
Answer: The solution is
Solve
Answer:
Check your solution.
Solve
Then graph the solution on a
number line.
The solution is
Graph the solution.
Place an open circle at
the left.
Answer:
Draw a line and an arrow to
Solve
number line.
Answer:
Then graph the solution on a
TOWING COMPANY A pickup truck is towing a
trailer that weighs 3,525 pounds. The maximum
towing capacity of the truck is 4,700 pounds.
Determine how much more weight can be added to
the trailer and still be towed by the truck.
Words
The phrase maximum capacity means less
than or equal to. So, the current weight being
towed plus any more weight must be less than
or equal to 4,700 pounds.
Variable
Inequality
current
weight
plus
3,525
+
must be
weight less than or 4,700
added
equal to pounds
w
4,700
Write the inequality.
Subtract 3,525 from
each side.
Simplify.
Answer: Up to 1,175 more pounds can be added
to the trailer.
SPORTS A weightlifter can lift up to 375 pounds. He
is currently lifting 255 pounds. Determine how much
more weight can be added and still be lifted by the
weightlifter.
Answer: Up to 120 more pounds can be added.
Example 1 Divide by a Positive Number
Example 2 Multiply by a Positive Number
Example 3 Multiply or Divide by a Negative Number
Example 4 Multiply or Divide by a Negative Number
Example 5 Solve a Two-Step Inequality
Solve
Check your solution.
Write the inequality.
Divide each side by 6.
Simplify.
Answer: The solution is
You can check this
solution by substituting numbers less than –5
into the inequality.
Solve
Answer:
Check your solution.
Solve
and check your solution. Then graph
the solution on a number line.
Write the inequality.
Multiply each side by 2.
Simplify.
The solution is
You can check this solution by
substituting 18 and a number greater than 18 into the
inequality.
Graph the solution,
Answer:
Solve
and check your solution. Then graph
the solution on a number line.
Answer:
Solve
Check your solution.
Write the inequality.
Multiply each side by –4 and
reverse the inequality symbol.
Simplify.
Answer: The solution is
You can check this
solution by replacing b in the original
inequality with –20 and a number greater
than –20.
Solve
Answer:
Check your solution.
Solve
number line.
Then graph the solution on a
Write the inequality.
Divide each side by –4 and
reverse the inequality symbol.
Check this result.
Graph the solution,
Answer:
Solve
number line.
Answer:
Then graph the solution on a
PACKAGES A box weighs 1 pound. It is filled with
books that weigh 2 pounds each. Jesse can carry at
most 20 pounds. Assuming space is not an issue, write
and solve an inequality to find how many books he can
put in the box and still carry it.
The phrase at most means less than or equal to.
of books he puts in the box. Then write
an inequality.
1
pound
1
plus
2 pounds
per book
2p
is less
than or
equal to
20
pounds
20
Write the inequality.
Subtract 1 from each side.
Simplify.
Divide each side by 2.
Simplify.
Answer: Since he can not put half a book in the box,
Jesse can put at most 9 books in the box.
PACKAGES A box weighs 2 pounds. It is filled with
toys that weigh 1 pound each. Danielle can carry at
most 30 pounds. Assuming space is not an issue, write
and solve an inequality to find how many toys she can
put in the box and still carry it.
Answer:
She can put at most 28 toys in
the box.
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