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Algebra 2 Interactive Chalkboard
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GLENCOE DIVISION
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, Ohio 43240
Lesson 1-1 Expressions and Formulas
Lesson 1-2 Properties of Real Numbers
Lesson 1-3 Solving Equations
Lesson 1-4 Solving Absolute Value Equations
Lesson 1-5 Solving Inequalities
Lesson 1-6 Solving Compound and Absolute
Value Inequalities
Example 1 Simplify an Expression
Example 2 Evaluate an Expression
Example 3 Expression Containing a Fraction Bar
Example 4 Use a Formula
Find the value of
First, subtract 2 from 7.
Then cube 5.
Multiply 125 by 3.
Subtract 375 from 384.
Finally, divide 9 by 3.
Answer: The value is 3.
Find the value of
Answer: 9
Evaluate
if
and
Replace s with 2 and
t with 3.4.
Find 22.
Subtract 3.4 from 4.
Multiply 3.4 and 0.6.
Subtract 2.04 from 2.
Answer: The value is –0.04.
Evaluate
Answer: –110
if
and
Evaluate
if
,
, and
Evaluate the numerator and
the denominator separately.
Multiply 40 by –2.
Simplify the numerator
and the denominator.
Then divide.
Answer: The value is –9.
Evaluate
Answer: –23
if
and
Find the area of a trapezoid with base lengths of 13
meters and 25 meters and a height of 8 meters.
Area of a trapezoid
Replace h with 8, b1 with
13, and b2 with 25.
Add 13 and 25.
Multiply 8 by
.
Multiply 4 and 38.
Answer: The area of the trapezoid is 152
square meters.
The formula for the volume V of a pyramid is
where B represents the area of the base and h is
the height of the pyramid. Find the volume of the
pyramid shown below.
Answer: 50 cm3
,
Example 1 Classify Numbers
Example 2 Identify Properties of Real Numbers
Example 3 Additive and Multiplicative Inverses
Example 4 Use the Distributive Property to Solve
a Problem
Example 5 Simplify an Expression
Name the sets of numbers to which
Answer: rationals (Q) and reals (R)
belongs.
Name the sets of numbers to which
belongs.
The bar over the 9 indicates that those digits repeat forever.
Answer: rationals (Q) and reals (R)
Name the sets of numbers to which
belongs.
lies between 2 and 3 so it is not a whole number.
Answer: irrationals (I) and reals (R)
Name the sets of numbers to which
belongs.
Answer: naturals (N), wholes (W), integers (Z),
rationals (Q) and reals (R)
Name the sets of numbers to which –23.3 belongs.
Answer: rationals (Q) and reals (R)
Name the sets of numbers to which each
number belongs.
a.
Answer: rationals (Q) and reals (R)
b.
Answer: rationals (Q) and reals (R)
c.
Answer: irrationals (I) and reals (R)
d.
Answer: naturals (N), wholes (W), integers (Z)
rationals (Q) and reals (R)
e. 32.1
Answer: rationals (Q) and reals (R)
Name the property illustrated by
.
The Additive Inverse Property says that a number plus its
opposite is 0.
Answer: Additive Inverse Property
Name the property illustrated by
.
The Distributive Property says that you multiply each term
within the parentheses by the first number.
Answer: Distributive Property
Name the property illustrated by each equation.
a.
Answer: Identity Property of Addition
b.
Answer: Inverse Property of Multiplication
Identify the additive inverse and multiplicative inverse
for –7.
Since –7 + 7 = 0, the additive inverse is 7.
Since
the multiplicative inverse is
Answer: The additive inverse is 7, and the multiplicative
inverse is
Identify the additive inverse and multiplicative inverse
for
Since
Since
.
the additive inverse is
the multiplicative inverse is
Answer: The additive inverse is
inverse is 3.
and the multiplicative
Identify the additive inverse and multiplicative inverse
for each number.
a. 5
Answer: additive: –5; multiplicative:
b.
Answer: additive:
multiplicative:
Postage Audrey went to a post office and bought
eight 34-cent stamps and eight 21-cent postcard
stamps. How much did Audrey spend altogether on
stamps?
There are two ways to find the total amount spent
on stamps.
Method 1
Multiply the price of each type of stamp by 8 and then add.
Method 2
Add the prices of both types of stamps and then multiply
the total by 8.
Answer: Audrey spent a total of $4.40 on stamps.
Notice that both methods result in the same answer.
Chocolate Joel went to the
grocery store and bought 3
plain chocolate candy bars for
$0.69 each and 3 chocolatepeanut butter candy bars for
$0.79 each. How much did
Joel spend altogether on
candy bars?
Answer: $4.44
Simplify
Distributive Property
Multiply.
Commutative Property (+)
Distributive Property
Answer:
Simplify.
Simplify
Answer:
.
Example 1 Verbal to Algebraic Expression
Example 2 Algebraic to Verbal Sentence
Example 3 Identify Properties of Equality
Example 4 Solve One-Step Equations
Example 5 Solve a Multi-Step Equation
Example 6 Solve for a Variable
Example 7 Apply Properties of Equality
Example 8 Write an Equation
Write an algebraic expression to represent 3 more
than a number.
Answer:
Write an algebraic expression to represent 6 times the
cube of a number.
Answer:
Write an algebraic expression to represent the square
of a number decreased by the product of 5 and the
same number.
Answer:
Write an algebraic expression to represent twice the
difference of a number and 6.
Answer:
Write an algebraic expression to represent each
verbal expression.
a. 6 more than a number
Answer:
b. 2 less than the cube of a number
Answer:
c. 10 decreased by the product of a number and 2
Answer:
d. 3 times the difference of a number and 7
Answer:
Write a verbal sentence to represent
Answer: The sum of 14 and 9 is 23.
.
Write a verbal sentence to represent
Answer: Six is equal to –5 plus a number.
.
Write a verbal sentence to represent
Answer: Seven times a number minus 2 is 19.
.
Write a verbal sentence to represent each equation.
a.
Answer: The difference between 10 and 3 is 7.
b.
Answer: Five is equal to the sum of 2 and a number.
c.
Answer: Three times a number plus 2 equals 11.
Name the property illustrated by the statement
if xy = 28 and x = 7, then 7y = 28.
Answer: Substitution Property of Equality
Name the property illustrated by the statement
.
Answer: Reflexive Property of Equality
Name the property illustrated by each statement.
a.
Answer: Symmetric Property of Equality
b.
Answer: Transitive Property of Equality
Solve
. Check your solution.
Original equation
Add 5.48 to each side.
Simplify.
Check:
Original equation
Substitute 5.5 for s.
Simplify.
Answer: The solution is 5.5.
Solve
. Check your solution.
Original equation
Multiply each side by
multiplicative inverse of
Simplify.
the
Check:
Original equation
Substitute 36 for t.
Simplify.
Answer: The solution is 36.
Solve each equation. Check your solution.
a.
Answer: –2
b.
Answer: 15
Solve
Original equation
Distributive and
Substitution Properties
Commutative, Distributive,
and Substitution Properties
Addition and
Substitution Properties
Division and
Substitution Properties
Answer: The solution is –19.
Solve
Answer: –6
Geometry The area of a trapezoid is
where A is the area, b1 is the length of one base, b2
is the length of the other base, and h is the height of
the trapezoid. Solve the formula for h.
Area of a trapezoid
Multiply each side by 2.
Simplify.
Divide each side by
Simplify.
.
Answer:
Geometry The perimeter of a rectangle is
where P is the perimeter, is the length, and w is the
width of the rectangle. Solve the formula for w.
w
Answer:
Multiple-Choice Test Item
what is the value of
A
B
C
D
Read the Test Item
You are asked to find the value of the expression 4g – 2.
Your first thought might be to find the value of g and then
evaluate the expression using this value. However, you
are not required to find the value of g. Instead, you can
use the Subtraction Property of Equality on the given
equation to find the value of 4g – 2.
Solve the Test Item
Original equation
Subtract 7 from each side.
Answer: B
Multiple-Choice Test Item
what is the value of
A 12
B 6
C –6
D –12
Answer: D
Home Improvement Carl wants to replace the 5
windows in the 2nd-story bedrooms of his home.
His neighbor Will is a carpenter and he has agreed
to help install them for $250. If Carl has budgeted
$1000 for the total cost, what is the maximum
amount he can spend on each window?
Explore Let c represent the cost of each window.
Plan
The number
the cost
the cost for a
the total
of windows times per window plus
carpenter
cost.
equals
5
c
+
250
=
1000
Solve
Original equation
Subtract 250 from each side.
Simplify.
Divide each side by 5.
Simplify.
Answer: Carl can afford to spend $150 on each window.
Examine The total cost to replace five windows at $150
each is 5(150) or $750. Add the $250 cost of
the carpenter to that, and the total bill to
replace the windows is 750 + 250 or $1000.
Thus, the answer is correct.
Home Improvement
Kelly wants to repair
the siding on her
house. Her contractor
will charge her $300
plus $150 per square foot of siding. How much
siding can she repair for $1500?
Answer: 8 ft2
Example 1 Evaluate an Expression with Absolute Value
Example 2 Solve an Absolute Value Equation
Example 3 No Solution
Example 4 One Solution
Evaluate
Replace x with 4.
Simplify –2(4) first.
Subtract 8 from 6.
Add.
Answer: The value is 4.7.
Evaluate
Answer: –13.7
Solve
Case 1
Check your solutions.
or
Case 2
Check:
Answer: The solutions are 5 or –11.
Thus, the solution set is
Solve
Answer:
Check your solutions.
Solve
Original equation
Subtract 5 from each side.
This sentence is never true.
Answer: The solution set is .
Solve
Answer: 
Solve
Case 1
Check your solutions.
or
Case 2
There appear to be two solutions, 11 or
Check:
Answer: Since
, the only solution is 11.
The solution set is {11}.
Solve
Answer: {6}
Example 1 Solve an Inequality Using Addition
or Subtraction
Example 2 Solve an Inequality Using Multiplication
or Division
Example 3 Solve a Multi-Step Inequality
Example 4 Write an Inequality
Solve
number line.
Graph the solution set on a
Original inequality
Subtract 4y from each side.
Simplify.
Subtract 2 from each side.
Simplify.
Rewrite with y first.
Answer: Any real number greater than –5 is a solution
of this inequality.
A circle means that this
point is not included in
the solution set.
Solve
number line.
Answer:
Graph the solution set on a
Solve
number line.
Graph the solution set on a
Original inequality
Divide each side by –0.3,
reversing the inequality symbol.
–40  p
p  –40
Simplify.
Rewrite with p first.
Answer: The solution set is
A dot means that this
point is included in the
solution set.
Solve
number line.
Answer:
Graph the solution set on a
Solve
Graph the solution set on a
number line.
Original inequality
Multiply each side by 2.
Add –x to each side.
Divide each side by –3,
reversing the inequality symbol.
Answer: The solution set is
graphed below.
and is
Solve
number line.
Answer:
Graph the solution set on a
Consumer Costs Alida has at most $10.50 to
spend at a convenience store. She buys a bag of
potato chips and a can of soda for $1.55. If gasoline
at this store costs $1.35 per gallon, how many
gallons of gasoline can Alida buy for her car, to the
nearest tenth of a gallon?
Explore Let
the number of gallons of gasoline that
Alida buys.
Plan
The total cost of the gasoline is 1.35g. The cost
of the chips and soda plus the total cost of the
gasoline must be less than or equal to $10.50.
Write an inequality.
The cost of
chips & soda
1.55
Solve
plus
the cost
of gasoline
is less than
or equal to
$10.50.
+
1.35g

10.50
Original inequality
Subtract 1.55
from each side.
Simplify.
Divide each side
by 1.35.
Simplify.
Answer: Alida can buy up to 6.6 gallons of gasoline for
her car.
Examine Since
is actually greater than 6.6, Alida
will have enough money if she gets no more
than 6.6 gallons of gasoline.
Rental Costs Jeb wants to rent a car for his
vacation. Value Cars rents cars for $25 per day plus
$0.25 per mile. How far can he drive for one day if
he wants to spend no more that $200 on car rental?
Answer: up to 700 miles
Example 1 Solve an “and” Compound Inequality
Example 2 Solve an “or” Compound Inequality
Example 3 Solve an Absolute Value Inequality (<)
Example 4 Solve an Absolute Value Inequality (>)
Example 5 Solve a Multi-Step Absolute Value Inequality
Example 6 Write an Absolute Value Inequality
Solve
Graph the solution set on a number line.
Method 1 Write the compound inequality using the word
and. Then solve each inequality.
and
Method 2 Solve both parts at the same time by adding 2
to each part. Then divide each part by 3.
Graph the solution set for each inequality and find
their intersection.
y4
Solve
Graph the solution set on a number line.
Answer:
Solve
or
Graph the solution set on a number line.
Solve each inequality separately.
or
Answer: The solution set is
Solve
Graph the solution set on a number line.
Answer:
Solve
Graph the solution set on a number line.
You can interpret
to mean that the distance
between d and 0 on a number line is less than 3 units. To
make
true, you must substitute numbers for d that
are fewer than 3 units from 0.
Notice that the graph of
is the same as the
graph of d > –3 and d < 3.
All of the numbers between –3 and 3 are less
than 3 units from 0.
Answer: The solution set is
Solve
Answer:
Graph the solution set on a number line.
Solve
Graph the solution set on a number line.
You can interpret
to mean that the distance
between d and 0 on a number line is greater than 3 units.
To make
true, you must substitute values for d
that are greater than 3 units from 0.
Notice that the graph of
is the same as the
graph of
All of the numbers not between –3 and 3 are greater
than 3 units from 0.
Answer: The solution set is
Solve
Answer:
Graph the solution set on a number line.
Solve
Graph the solution set on a number line.
is equivalent to
Solve each inequality.
or
Answer: The solution set is
.
Solve
Graph the solution set on a number line.
Answer:
Housing According to a recent survey, the average
monthly rent for a one-bedroom apartment in one city
is $750. However, the actual rent for any given onebedroom apartment might vary as much as $250 from
the average.
Write an absolute value inequality to describe this
situation.
Let
the actual monthly rent.
The rent for an apartment
can differ from the average
Answer:
by as much as
$250.

250
Solve the inequality
monthly rent.
to find the range of
Rewrite the absolute value inequality as a compound
inequality. Then solve for r.
–r
–r
r
Answer: The solution set is
The actual rent falls between
$500 and $1000, inclusive.
Health The average birth weight of a newborn baby is
7 pounds. However, this weight can vary by as much
as 4.5 pounds.
a. Write an absolute value inequality to describe
this situation.
Answer:
b. Solve the inequality to find the range of birth weights
for newborn babies.
Answer:
The birth weight of a
newborn baby will fall between 2.5 pounds and 11.5
pounds, inclusive.
Explore online information about the
information introduced in this chapter.
Click on the Connect button to launch your browser
and go to the Algebra 2 Web site. At this site, you
will find extra examples for each lesson in the
Student Edition of your textbook. When you finish
exploring, exit the browser program to return to this
presentation. If you experience difficulty connecting
to the Web site, manually launch your Web browser
and go to www.algebra2.com/extra_examples.
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