Download Evaluate Expressions

1.1
Evaluate Expressions
Goal p Evaluate algebraic expressions and use exponents.
Your Notes
VOCABULARY
Variable
An algebraic
expression is also
called a variable
expression.
Algebraic expression
Evaluating an expression
Power
Base
Exponent
ALGEBRAIC EXPRESSIONS
Algebraic
Expression
Meaning
7t
7 times t
x
20
}
Operation
Division
y28
12 1 a
Copyright © Holt McDougal. All rights reserved.
Lesson 1.1 • Algebra 1 Notetaking Guide
1
1.1
Evaluate Expressions
Goal p Evaluate algebraic expressions and use exponents.
Your Notes
VOCABULARY
Variable A letter used to represent one or more
numbers
An algebraic
expression is also
called a variable
expression.
Algebraic expression A collection of numbers,
variables, and operations
Evaluating an expression Substituting a number
for the variable, performing the operation(s), and
simplifying the result if necessary
Power An expression that represents repeated
multiplication of the same factor
Base The number or expression that is used as a
factor in repeated multiplication
Exponent A number or expression that represents
the number of times the base is used as a factor
ALGEBRAIC EXPRESSIONS
Algebraic
Expression
Meaning
7t
7 times t
Operation
Multiplication
}
x
20
x divided by 20
y28
y minus 8
Subtraction
12 1 a
12 plus a
Addition
Copyright © Holt McDougal. All rights reserved.
Division
Lesson 1.1 • Algebra 1 Notetaking Guide
1
Your Notes
To evaluate
an expression,
substitute a
number for the
variable, perform
the operation(s),
and simplify.
Evaluate algebraic expressions
Example 1
Evaluate the expression when n 5 4.
a. 11 3 n 5 11 3
Substitute
for n.
.
5
12
12
5}
b. }
n
Substitute
for n.
.
5
c. n 2 3 5
Substitute
23
for n.
.
5
Checkpoint Evaluate the expression when y 5 8.
1. 7y
2. y 4 2
Example 2
3. 10 2 y
4. y 1 6
Read and write powers
Write the power in words and as a product.
Power
Product
a. 121
twelve to the
power
b. 23
two to the
power,
or two
1
c. 1 }
42
d. a4
2
Words
Lesson 1.1 • Algebra 1 Notetaking Guide
2
one fourth to the
power, or
one fourth
a to the
power
Copyright © Holt McDougal. All rights reserved.
Your Notes
To evaluate
an expression,
substitute a
number for the
variable, perform
the operation(s),
and simplify.
Evaluate algebraic expressions
Example 1
Evaluate the expression when n 5 4.
a. 11 3 n 5 11 3 4
Substitute 4 for n.
44
5
Multiply .
12
12
5}
b. }
n
4
Substitute 4 for n.
5 3
Divide
c. n 2 3 5 4 2 3
.
Substitute 4 for n.
5 1
Subtract .
Checkpoint Evaluate the expression when y 5 8.
1. 7y
56
Example 2
2. y 4 2
3. 10 2 y
4
4. y 1 6
2
14
Read and write powers
Write the power in words and as a product.
Power
Product
a. 121
twelve to the
first power
b. 23
two to the
third power,
or two cubed
1 2
c. }
4
1 2
d. a4
2
Words
Lesson 1.1 • Algebra 1 Notetaking Guide
one fourth to the
second power, or
one fourth squared
a to the fourth
power
12
2p2p2
1
4
1
4
}p}
apapapa
Copyright © Holt McDougal. All rights reserved.
Your Notes
Checkpoint Write the power in words and as a product.
2
1
6. 1 }
32
5. 75
7. (1.4)3
Evaluate powers
Example 3
Evaluate the expression.
a. y 3 when y 5 3
Solution
a. y 3 5
b. a5 when a 5 1.2
3
Substitute
for y.
5
.
5
.
b. a5 5
5
Substitute
for a.
5
.
5
.
Checkpoint Evaluate the expression.
8. t 2 when t 5 3
Homework
Copyright © Holt McDougal. All rights reserved.
9. m5 when
1
m5}
2
10. x 3 when
x54
Lesson 1.1 • Algebra 1 Notetaking Guide
3
Your Notes
Checkpoint Write the power in words and as a product.
2
5. 75
1
6. 1 }
32
7. (1.4)3
seven to the
fifth power
one third to the
second power,
or one third
squared
one and four
tenths to the
third power, or
one and four
tenths cubed
7p7p7p7p7
1
3
1
3
}p}
(1.4)(1.4)(1.4)
Evaluate powers
Example 3
Evaluate the expression.
a. y 3 when y 5 3
Solution
a. y 3 5 3
b. a5 when a 5 1.2
3
Substitute 3 for y.
5 3 p 3 p 3
Write factors
.
5 27
Multiply
.
b. a5 5 1.2
5
Substitute 1.2 for a.
5 (1.2)(1.2)(1.2)(1.2)(1.2)
Write factors
.
5 2.48832
Multiply
.
Checkpoint Evaluate the expression.
8. t 2 when t 5 3
Homework
9
9. m5 when
1
m5}
2
x54
1
32
64
}
Copyright © Holt McDougal. All rights reserved.
10. x 3 when
Lesson 1.1 • Algebra 1 Notetaking Guide
3
1.2
Apply Order of Operations
Goal p Use the order of operations to evaluate expressions.
Your Notes
VOCABULARY
Order of Operations
ORDER OF OPERATIONS
To evaluate an expression involving more than one operation,
use the following steps.
Step 1 Evaluate expressions inside
.
Step 2 Evaluate
Step 3
.
and divide from left to right.
Step 4 Add and
Example 1
from left to right.
Evaluate Expressions
Evaluate the expression 30 3 2 4 22 2 5.
Solution
Step 1
There are no grouping symbols, so go to Step 2.
Step 2
30 3 2 4 22 2 5 5 30 3 2 4
25
power.
Step 3
30 3 2 4
255
5
4
25
25
.
.
Step 4
255
4
Lesson 1.2 • Algebra 1 Notetaking Guide
.
Copyright © Holt McDougal. All rights reserved.
1.2
Apply Order of Operations
Goal p Use the order of operations to evaluate expressions.
Your Notes
VOCABULARY
Order of Operations The rules established to
evaluate an expression involving more than one
operation
ORDER OF OPERATIONS
To evaluate an expression involving more than one operation,
use the following steps.
Step 1 Evaluate expressions inside grouping
symbols .
Step 2 Evaluate powers .
Step 3 Multiply and divide from left to right.
Step 4 Add and subtract from left to right.
Example 1
Evaluate Expressions
Evaluate the expression 30 3 2 4 22 2 5.
Solution
Step 1
There are no grouping symbols, so go to Step 2.
Step 2
30 3 2 4 22 2 5 5 30 3 2 4 4 2 5
Evaluate
power.
Step 3
30 3 2 4 4 2 5 5 60 4 4 2 5
5 15 2 5
Multiply .
Divide
.
Subtract
.
Step 4
15 2 5 5 10
4
Lesson 1.2 • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Checkpoint Evaluate the expression.
1. 10 1 32
2. 16 2 23 1 4
3. 28 4 22 1 1
4. 4 p 52 1 4
Example 2
Evaluate expressions with grouping symbols
Evaluate the expression.
Grouping symbols
such as parentheses
( ) and brackets
[ ] indicate that
operations inside the
grouping symbols
should be performed
first.
a. 6(9 1 3) 5 6(
within
parentheses.
)
.
5
b. 50 2 (32 1 1) 5 50 2 (
1 1)
power.
5 50 2 (
within
parentheses.
)
.
5
c. 3[5 1 (52 1 5)] 5 3[5 1 (
1 5)]
power.
5 3[5 1 (
5 3[
5
Copyright © Holt McDougal. All rights reserved.
]
)]
within
parentheses.
within
brackets.
.
Lesson 1.2 • Algebra 1 Notetaking Guide
5
Your Notes
Checkpoint Evaluate the expression.
1. 10 1 32
2. 16 2 23 1 4
19
12
4. 4 p 52 1 4
3. 28 4 22 1 1
8
Example 2
104
Evaluate expressions with grouping symbols
Evaluate the expression.
Grouping symbols
such as parentheses
( ) and brackets
[ ] indicate that
operations inside the
grouping symbols
should be performed
first.
a. 6(9 1 3) 5 6( 12 )
Add within
parentheses.
5 72
Multiply .
b. 50 2 (32 1 1) 5 50 2 ( 9 1 1)
5 50 2 ( 10 )
5 40
Add within
parentheses.
Subtract .
c. 3[5 1 (52 1 5)] 5 3[5 1 ( 25 1 5)]
Evaluate
power.
5 3[5 1 ( 30 )]
Add within
parentheses.
5 3[ 35 ]
Add within
brackets.
5 105
Copyright © Holt McDougal. All rights reserved.
Evaluate
power.
Multiply .
Lesson 1.2 • Algebra 1 Notetaking Guide
5
Your Notes
Checkpoint Evaluate the expression.
5. 6(3 1 32)
Example 3
6. 2[(10 2 4) 4 3]
Evaluate an algebraic expression
12k
Evaluate the expression }
when k 5 2.
2
3(k 1 4)
A fraction bar can
act as a grouping
symbol. Evaluate
the numerator and
denominator before
dividing.
Solution
12k
3(k 1 4)
} 5
2
5
5
12(
3(
)
2
12(
Substitute
for k.
1 4)
)
3(
1 4)
12(
)
3(
)
power.
within parentheses.
5
.
5
.
Checkpoint Evaluate the expression when x 5 3.
Homework
6
7. x3 2 5
Lesson 1.2 • Algebra 1 Notetaking Guide
6x 1 2
8. }
x17
Copyright © Holt McDougal. All rights reserved.
Your Notes
Checkpoint Evaluate the expression.
5. 6(3 1 32)
6. 2[(10 2 4) 4 3]
72
4
Evaluate an algebraic expression
Example 3
12k
Evaluate the expression }
when k 5 2.
2
3(k 1 4)
A fraction bar can
act as a grouping
symbol. Evaluate
the numerator and
denominator before
dividing.
Solution
12k
3(k 1 4)
} 5
2
5
5
5
12( 2 )
3( 2
2
1 4)
12( 2 )
3( 4
3( 8 )
24
5 1
Evaluate power.
1 4)
12( 2 )
24
Substitute 2 for k.
Add within parentheses.
Multiply
.
Divide
.
Checkpoint Evaluate the expression when x 5 3.
Homework
6
7. x3 2 5
22
Lesson 1.2 • Algebra 1 Notetaking Guide
6x 1 2
8. }
x17
2
Copyright © Holt McDougal. All rights reserved.
Focus On
Reasoning
Analyze Always, Sometimes,
Never Statements
Use after Lesson 1.2
Goal
Your Notes
p Determine whether statements are always,
sometimes, or never true.
Example 1
Determine the truth of a statement
Tell whether the statement is always, sometimes,
or never true.
For a given whole number, n, the product 3n represents
an odd whole number.
Solution
Test the expression for some values for n. When n is 1,
3n 5
, an
number. When n is 2, 3n 5
,
an
number. The statement is
true.
Example 2
Determine the truth of a statement
Tell whether the statement is always, sometimes,
or never true.
For a given whole number, n, the expression 4n 1 2
represents an even whole number.
Solution
Test the expression for some values of n. When n is 1,
, an
number. When n is 2,
, an
number. Will this happen
for any value of n?
Check whether dividing the expression by
results
in a whole number.
(4n 1 2) 4
5 (4n 4
) 1 (2 4
)
5
Because n is a whole number,
be a whole number. So, (4n 1 2)
and the original statement is
Copyright © Holt McDougal. All rights reserved.
will
divisible by
true.
1.2 Focus On Reasoning • Algebra 1 Notetaking Guide
,
7
Focus On
Reasoning
Analyze Always, Sometimes,
Never Statements
Use after Lesson 1.2
Goal
Your Notes
p Determine whether statements are always,
sometimes, or never true.
Example 1
Determine the truth of a statement
Tell whether the statement is always, sometimes,
or never true.
For a given whole number, n, the product 3n represents
an odd whole number.
Solution
Test the expression for some values for n. When n is 1,
3n 5 3 , an odd number. When n is 2, 3n 5 6 ,
an even number. The statement is sometimes
true.
Example 2
Determine the truth of a statement
Tell whether the statement is always, sometimes,
or never true.
For a given whole number, n, the expression 4n 1 2
represents an even whole number.
Solution
Test the expression for some values of n. When n is 1,
4n 1 2 5 6 , an even number. When n is 2,
4n 1 2 5 10 , an even number. Will this happen
for any value of n?
Check whether dividing the expression by 2 results
in a whole number.
(4n 1 2) 4 2 5 (4n 4 2 ) 1 (2 4 2 )
5 2n 1 1
Because n is a whole number, 2n 1 1 will always
be a whole number. So, (4n 1 2) is divisible by 2 ,
and the original statement is always true.
Copyright © Holt McDougal. All rights reserved.
1.2 Focus On Reasoning • Algebra 1 Notetaking Guide
7
Your Notes
Example 3
Determine the truth of a statement
Tell whether the statement is always, sometimes,
or never true.
For a given whole number, n, the expression (2n)2 1 1
represents an even whole number.
Solution
Test the expression for some values of n. When n is 1,
, an
number. When n is 2,
,
. Will this
happen for any value of n?
Check whether
in a
((2n)2 1 1) 4
results
.
5 ((2n)2 4
5(
) 1 (1 4
) 1 (1 4
4
)
)
1
5
1}
Because n is a whole number, the first term
will
be a whole number. This means that
the sum
(2n)2 1 1
statement is
be a whole number. So,
divisible by
, and the original
true.
Checkpoint Tell whether the statement is always,
sometimes, or never true..
Homework
1. For a given whole number, n, the expression
represents an even whole number.
2. For a given odd whole number, n, the expression
n2 1 2 represents an even whole number.
8
1.2 Focus On Reasoning • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Example 3
Determine the truth of a statement
Tell whether the statement is always, sometimes,
or never true.
For a given whole number, n, the expression (2n)2 1 1
represents an even whole number.
Solution
Test the expression for some values of n. When n is 1,
(2n)2 1 1 5 5 , an odd number. When n is 2,
(2n)2 1 1 5 17 , an odd number . Will this
happen for any value of n?
Check whether dividing the expression by 2
in a whole number .
results
((2n)2 1 1) 4 2 5 ((2n)2 4 2 ) 1 (1 4 2 )
5 ( 4n2 4 2
) 1 (1 4 2 )
1
5 2n2 1 }
2
Because n is a whole number, the first term 2n2
will always be a whole number. This means that
1
the sum 2n2 1 }
(2n)2 1 1
2
will not be a whole number. So,
is not divisible by 2 , and the original
statement is
never
true.
Checkpoint Tell whether the statement is always,
sometimes, or never true..
Homework
1. For a given whole number, n, the expression
represents an even whole number.
sometimes
2. For a given odd whole number, n, the expression
n2 1 2 represents an even whole number.
never
8
1.2 Focus On Reasoning • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
1.3
Write Expressions
Goal p Translate verbal phrases into expressions.
Your Notes
VOCABULARY
Verbal model
Rate
Unit rate
TRANSLATING VERBAL PHRASES
Operation
Verbal Phrase
Expression
Addition
The
of 3 and
a number n
A number x
Subtraction
Order is important
when writing
subtraction
and division
expressions.
10
The
of 7
and a number a
Twelve
a number x
Multiplication
Five
than
a number y
The
of 2
and a number n
Division
The
of a
number a and 6
Eight
a number y
Copyright © Holt McDougal. All rights reserved.
into
Lesson 1.3 • Algebra 1 Notetaking Guide
9
1.3
Write Expressions
Goal p Translate verbal phrases into expressions.
Your Notes
VOCABULARY
Verbal model An expression that describes a
problem using words as labels and using math
symbols to relate the words
Rate A fraction that compares two quantities
measured in different units
Unit rate A rate per one given unit
TRANSLATING VERBAL PHRASES
Operation
Verbal Phrase
Addition
The sum of 3 and
a number n
31n
A number x plus 10
x 1 10
The difference of 7
72a
Subtraction
Order is important
when writing
subtraction
and division
expressions.
Expression
and a number a
Multiplication
Division
Copyright © Holt McDougal. All rights reserved.
Twelve less than
a number x
x 2 12
Five times a number y
5y
The product of 2
and a number n
2n
The quotient of a
number a and 6
}
Eight divided into
a number y
}
a
6
y
8
Lesson 1.3 • Algebra 1 Notetaking Guide
9
Your Notes
Example 1
Translate verbal phrases into expressions
Translate the verbal phrase into an expression.
The words “the
quantity” tell you
what to group when
translating verbal
phrases.
Verbal Phrase
Expression
a. 6 less than the quantity
8 times a number x
b. 2 times the sum of 5
and a number a
c. The difference of 17 and
the cube of a number n
Checkpoint Translate the verbal phrase into an
expression.
1. The product of 5 and the quantity 12 plus a number n
2. The quotient of 10 and the quantity a number x
minus 3
Example 2
Use a verbal model to write an expression
Food Drive You and three friends are collecting canned
food for a food drive. You each collect the same number
of cans. Write an expression for the total number of cans
collected.
Solution
Step 1 Write a verbal model. Amount 3 Number of
of cans
Step 2 Translate the verbal
model into an
algebraic expression.
3
An expression that represents the total number of cans
is
.
10 Lesson 1.3 • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Example 1
Translate verbal phrases into expressions
Translate the verbal phrase into an expression.
The words “the
quantity” tell you
what to group when
translating verbal
phrases.
Verbal Phrase
Expression
a. 6 less than the quantity
8 times a number x
8x 2 6
b. 2 times the sum of 5
and a number a
2(5 1 a)
c. The difference of 17 and
the cube of a number n
17 2 n3
Checkpoint Translate the verbal phrase into an
expression.
1. The product of 5 and the quantity 12 plus a number n
5(12 1 n)
2. The quotient of 10 and the quantity a number x
minus 3
10
x23
}
Example 2
Use a verbal model to write an expression
Food Drive You and three friends are collecting canned
food for a food drive. You each collect the same number
of cans. Write an expression for the total number of cans
collected.
Solution
Step 1 Write a verbal model. Amount 3 Number of
of cans
people
Step 2 Translate the verbal
model into an
algebraic expression.
c
3
4
An expression that represents the total number of cans
is c 3 4 .
10 Lesson 1.3 • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Checkpoint Complete the following exercise.
3. In Example 2, suppose that the total number of cans
collected are distributed equally to 2 food banks.
Write an expression that represents the number of
cans each food bank receives.
Find a unit rate
Example 3
A construction crew digs a tunnel 430 feet long in
3 days. Find their unit rate in miles per week.
Solution
} 5
days
Use unit analysis,
also called
dimensional
analysis, to check
that the expression
produces an answer
in miles per week.
5280
days
1
mile
5}
1
Check that the expression produces the units you expect.
mile
days
mile
}•}• }5 }
days
Their unit rate is
Homework
days
1 mile
} •}
• }
mile per
.
Checkpoint Find the unit rate.
4. A worker earns a nickel every eight seconds. What is
his rate in dollars per hour?
Copyright © Holt McDougal. All rights reserved.
Lesson 1.3 • Algebra 1 Notetaking Guide
11
Your Notes
Checkpoint Complete the following exercise.
3. In Example 2, suppose that the total number of cans
collected are distributed equally to 2 food banks.
Write an expression that represents the number of
cans each food bank receives.
c34
}
2
Find a unit rate
Example 3
A construction crew digs a tunnel 430 feet long in
3 days. Find their unit rate in miles per week.
Solution
430 feet
} 5
3 days
Use unit analysis,
also called
dimensional
analysis, to check
that the expression
produces an answer
in miles per week.
430 feet
7
1 week
days
1 mile
} •}
• }
5280 feet
3 days
0.19 mile
5}
1 week
Check that the expression produces the units you expect.
feet
mile
days
mile
}5 }
}•}
week
feet • week
days
Their unit rate is 0.19 mile per week .
Homework
Checkpoint Find the unit rate.
4. A worker earns a nickel every eight seconds. What is
his rate in dollars per hour?
$22.50/hr
Copyright © Holt McDougal. All rights reserved.
Lesson 1.3 • Algebra 1 Notetaking Guide
11
1.4
Write Equations and
Inequalities
Goal
Your Notes
p Translate verbal sentences into equations or
inequalities.
VOCABULARY
Open sentence
Equation
Inequality
Solution of an equation
Solution of an inequality
EXPRESSING OPEN SENTENCES
Symbol
Meaning
a5b
a is
a<b
a is
a≤b
a is
or
a>b
a is
a≥b
a is
or
12 Lesson 1.4 • Algebra 1 Notetaking Guide
Associated Words
to b
a is the
b
a is
than
to b
a is
a is
than b
b
than
to b
a is
a is
a is
as b
than b
b,
than b
b,
than b
Copyright © Holt McDougal. All rights reserved.
1.4
Write Equations and
Inequalities
Goal
Your Notes
p Translate verbal sentences into equations or
inequalities.
VOCABULARY
Open sentence A mathematical statement that
contains two expressions and a symbol that
compares them
Equation An open sentence that contains the
symbol 5
Inequality An open sentence that contains one of
the symbols <, ≤, >, or ≥
Solution of an equation A number that when
substituted for the variable in an equation results
in a true statement
Solution of an inequality A number that when
substituted for the variable in an inequality results
in a true statement
EXPRESSING OPEN SENTENCES
Symbol
Meaning
Associated Words
a5b
a is equal to b
a is the same as b
a<b
a is less than b
a is fewer than b
a≤b
a is less than
or equal to b
a is at most b,
a is no more
than b
a>b
a is greater than b
a is more than b
a≥b
a is greater than
or equal to b
a is at least b,
a is no less than b
12 Lesson 1.4 • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Example 1
Write equations and inequalities
Write an equation or an inequality.
Verbal Sentence
Sometimes two
inequalities are
combined. For
example, the
inequalities a < b
and b < c can be
combined to form
the inequality
a < b < c.
Equation or Inequality
a. The sum of three times a
number a and 4 is 25.
b. The quotient of a number
x and 4 is fewer than 10.
c. A number n is greater
than 6 and less than 12.
Example 2
Check possible solutions
Check whether 2 is a solution of the equation or
inequality.
Equation or
Inequality
a. 7x 2 8 5 9
b. 4 1 5y < 18
c. 6n 2 9 ≥ 2
Substitute
Conclusion
7(2) 2 8 0 9
4 1 5(2) ?
< 18
6(2) 2 9 ?
≥ 2
a solution.
a solution.
a solution.
Checkpoint Check whether the given number is a
solution of the equation or inequality.
1. 6r 1 1 5 25
r54
Copyright © Holt McDougal. All rights reserved.
2. x2 2 5 > 10
x55
3. 7a ≤ 21
a56
Lesson 1.4 • Algebra 1 Notetaking Guide
13
Your Notes
Example 1
Write equations and inequalities
Write an equation or an inequality.
Verbal Sentence
Sometimes two
inequalities are
combined. For
example, the
inequalities a < b
and b < c can be
combined to form
the inequality
a < b < c.
Equation or Inequality
a. The sum of three times a
number a and 4 is 25.
3a 1 4 5 25
b. The quotient of a number
x and 4 is fewer than 10.
} < 10
c. A number n is greater
than 6 and less than 12.
6 < n < 12
Example 2
x
4
Check possible solutions
Check whether 2 is a solution of the equation or
inequality.
Equation or
Inequality
a. 7x 2 8 5 9
b. 4 1 5y < 18
c. 6n 2 9 ≥ 2
Substitute
7(2) 2 8 0 9
4 1 5(2) ?
< 18
6(2) 2 9 ?
≥ 2
Conclusion
6Þ9
2 is not a solution.
14 < 18
2 is
a solution.
3≥2
2 is
a solution.
Checkpoint Check whether the given number is a
solution of the equation or inequality.
1. 6r 1 1 5 25
2. x2 2 5 > 10
3. 7a ≤ 21
r54
x55
a56
solution
solution
not a
solution
Copyright © Holt McDougal. All rights reserved.
Lesson 1.4 • Algebra 1 Notetaking Guide
13
Your Notes
Example 3
Use mental math to solve an equation
Solve the equation using mental math.
a. n 1 6 5 11
b. 18 2 x 5 10
c. 7a 5 56
d. } 5 3
11
Solution
Equation
Think of an equation
as a question
when solving using
mental math.
b
Think
Solution Check
a. n 1 6 5 11 What number
plus 6 equals
11?
1 6 5 11
b. 18 2 x 5 10
18 2
c. 7a 5 56
7(
b
53
d. }
11
5 10
) 5 56
}53
11
Checkpoint Solve the equation using mental math.
4. x 1 9 5 14
5. 5t 2 4 5 11
y
4
6. } 5 15
Homework
14 Lesson 1.4 • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Example 3
Use mental math to solve an equation
Solve the equation using mental math.
a. n 1 6 5 11
b. 18 2 x 5 10
c. 7a 5 56
d. } 5 3
11
Solution
Equation
Think of an equation
as a question
when solving using
mental math.
b
Think
Solution Check
a. n 1 6 5 11 What number
plus 6 equals
11?
5
5 1 6 5 11
b. 18 2 x 5 10 18 minus what
number equals
8
18 2 8 5 10
8
7( 8 ) 5 56
33
}53
10?
c. 7a 5 56
7 times
what number
equals 56?
b
53
d. }
11
What number
divided by 11
33
11
equals 3?
Checkpoint Solve the equation using mental math.
4. x 1 9 5 14
5
5. 5t 2 4 5 11
3
y
4
6. } 5 15
60
Homework
14 Lesson 1.4 • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
1.5
Use a Problem Solving Plan
Goal p Use a problem solving plan to solve problems.
Your Notes
VOCABULARY
Formula
A PROBLEM SOLVING PLAN
Use the following four-step plan to solve a problem.
Step 1
Read the problem
carefully. Identify what you want to know and
what you want to find out.
Step 2
Decide on an
approach to solving the problem.
Step 3
Carry out your plan.
Try a new approach if the first one isn’t successful.
Step 4
Check that your
answer is reasonable.
Example 1
Read a problem and make a plan
You have $7 to buy orange juice and bagels at the store.
A container of juice costs $1.25 and a bagel costs $.75.
If you buy two containers of juice, how many bagels can
you buy?
Solution
Step 1
What do you know? You
know how much money you have and the price of
a
and a container of juice.
What do you want to find out? You want to find out the
number of
you can buy.
Step 2
Copyright © Holt McDougal. All rights reserved.
Use what you know to write
a
that represents what you want
to find out. Then write an
and solve it.
Lesson 1.5 • Algebra 1 Notetaking Guide
15
1.5
Use a Problem Solving Plan
Goal p Use a problem solving plan to solve problems.
Your Notes
VOCABULARY
Formula An equation that relates two or more
quantities
A PROBLEM SOLVING PLAN
Use the following four-step plan to solve a problem.
Step 1 Read and Understand Read the problem
carefully. Identify what you want to know and
what you want to find out.
Step 2 Make a Plan
Decide on an
approach to solving the problem.
Step 3 Solve the Problem
Carry out your plan.
Try a new approach if the first one isn’t successful.
Step 4 Look Back
answer is reasonable.
Example 1
Check that your
Read a problem and make a plan
You have $7 to buy orange juice and bagels at the store.
A container of juice costs $1.25 and a bagel costs $.75.
If you buy two containers of juice, how many bagels can
you buy?
Solution
Step 1 Read and Understand What do you know? You
know how much money you have and the price of
a bagel and a container of juice.
What do you want to find out? You want to find out the
number of bagels you can buy.
Step 2 Make a Plan Use what you know to write
a verbal model that represents what you want
to find out. Then write an equation and solve it.
Copyright © Holt McDougal. All rights reserved.
Lesson 1.5 • Algebra 1 Notetaking Guide
15
Your Notes
Solve a problem and look back
Example 2
Solve the problem in Example 1 by carrying out the
plan. Then check your answer.
Solution
Step 3
Write a verbal model. Then
write an equation. Let b be the number of bagels you buy.
Price of
Number
Price of Number
Cost
juice
of
bagel
of
(in dollars)
(in dollars) containers (in dollars) bagels
p
p
1
b
5
The equation is
1
b5
. One way to solve
the equation is to use the strategy guess, check, and revise.
Guess an even number that is easily multiplied by
Try 4.
Write equation.
b5
1
Substitute 4 for b.
(4) 0
1
.
Simplify; 4
check.
Because
Try 6.
, try an even number
Write equation.
b5
1
Substitute 6 for b.
(6) 0
1
4.
Simplify.
For
juice.
you can buy
Step 4
containers of
Each additional bagel you buy adds
you pay for the juice. Make a table.
to the
Bagels
bagels and
0
1
2
3
4
5
6
Total Cost
The total cost is
when you buy
bagels and
containers of juice. The answer in step 3 is
.
16 Lesson 1.5 • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Solve a problem and look back
Example 2
Solve the problem in Example 1 by carrying out the
plan. Then check your answer.
Solution
Step 3 Solve the Problem Write a verbal model. Then
write an equation. Let b be the number of bagels you buy.
Price of
Number
Price of Number
Cost
juice
of
bagel
of
(in dollars)
(in dollars) containers (in dollars) bagels
1.25
p
2
1
0.75
p
b
7
5
The equation is 2.5 1 0.75 b 5 7 . One way to solve
the equation is to use the strategy guess, check, and revise.
Guess an even number that is easily multiplied by 0.75 .
Try 4.
2.5 1 0.75 b 5 7
Write equation.
2.5 1 0.75 (4) 0 7
Substitute 4 for b.
5.5 Þ 7
Simplify; 4 does not
check.
Because 5.5 < 7 , try an even number greater than 4.
Try 6.
2.5 1 0.75 b 5 7
Write equation.
2.5 1 0.75 (6) 0 7
Substitute 6 for b.
7 5 7
Simplify.
For $7 you can buy 6 bagels and 2 containers of
juice.
Step 4 Look Back Each additional bagel you buy adds
$.75 to the $2.50 you pay for the juice. Make a table.
Bagels
Total Cost
0
1
2.50 3.25
2
3
4
5
6
4
4.75
5.5
6.25
7
The total cost is $7 when you buy 6 bagels and 2
containers of juice. The answer in step 3 is correct .
16 Lesson 1.5 • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Checkpoint Complete the following exercise.
1. Suppose in Example 1 that you have $12 and you
decide to buy three containers of juice. How many
bagels can you buy?
FORMULA REVIEW
Temperature
5
C5}
(F 2 32), where F 5
9
and C 5
Simple interest
I 5 Prt, where I 5
,P5
r5
(as a decimal), and t 5
,
Distance traveled
d 5 rt, where d 5
and t 5
,r5
,
Profit
P 5 I 2 E, where P 5
E5
,I5
, and
Checkpoint Complete the following exercise.
Homework
2. In Example 1, the store where you bought the
juice and bagels had an income of $7 from your
purchase. The profit the store made from your
purchase is $2.50. Find the store’s expense for the
juice and bagels.
Copyright © Holt McDougal. All rights reserved.
Lesson 1.5 • Algebra 1 Notetaking Guide
17
Your Notes
Checkpoint Complete the following exercise.
1. Suppose in Example 1 that you have $12 and you
decide to buy three containers of juice. How many
bagels can you buy?
11 bagels
FORMULA REVIEW
Temperature
5
C5}
(F 2 32), where F 5 degrees Fahrenheit
9
and C 5 degrees Celsius
Simple interest
I 5 Prt, where I 5 interest , P 5 principal ,
r 5 interest rate (as a decimal), and t 5 time
Distance traveled
d 5 rt, where d 5 distance traveled , r 5 rate ,
and t 5 time
Profit
P 5 I 2 E, where P 5 profit , I 5 income , and
E 5 expenses
Checkpoint Complete the following exercise.
Homework
2. In Example 1, the store where you bought the
juice and bagels had an income of $7 from your
purchase. The profit the store made from your
purchase is $2.50. Find the store’s expense for the
juice and bagels.
$4.50
Copyright © Holt McDougal. All rights reserved.
Lesson 1.5 • Algebra 1 Notetaking Guide
17
1.6
Represent Functions
as Rules and Tables
Goal
Your Notes
p Represent functions as rules and as tables.
VOCABULARY
Function
Domain
Range
Independent variable
Dependent variable
Example 1
Identify the domain and range of a function
The input-output table shows temperatures over various
increments of time. Identify the domain and range of
the function.
Input (hours)
0
2
4
6
Output (°C)
24
27
30
33
Solution
Domain:
Range:
18 Lesson 1.6 • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
1.6
Represent Functions
as Rules and Tables
Goal
Your Notes
p Represent functions as rules and as tables.
VOCABULARY
Function A rule that establishes a relationship
between two quantities, called the input and the
output. For each input, there is exactly one output
Domain The collection of all input values
Range The collection of all output values
Independent variable The input variable
Dependent variable The output variable
Example 1
Identify the domain and range of a function
The input-output table shows temperatures over various
increments of time. Identify the domain and range of
the function.
Input (hours)
0
2
4
6
Output (°C)
24
27
30
33
Solution
Domain:
0, 2, 4, 6
Range:
24, 27, 30, 33
18 Lesson 1.6 • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Checkpoint Identify the domain and range of the
function.
1. Input
Output
Example 2
4
7
11
13
10
20
35
45
Identify a function
Tell whether the pairing is a function. Explain your
reasoning.
Mapping diagrams
are often used to
represent functions.
Take note of the
pairings to make your
decision.
Solution
a. Input
4
8
2
b. Input Output
Output
1
2
3
2
2
2
4
3
6
4
8
Checkpoint Tell whether the pairing is a function.
2. Input
Output
3. Input
Output
Copyright © Holt McDougal. All rights reserved.
5
5
10
15
3
4
6
8
0
4
12
20
3
5
9
13
Lesson 1.6 • Algebra 1 Notetaking Guide
19
Your Notes
Checkpoint Identify the domain and range of the
function.
1. Input
Output
4
7
11
13
10
20
35
45
Domain: 4, 7, 11, 13
Range: 10, 20, 35, 45
Example 2
Identify a function
Tell whether the pairing is a function. Explain your
reasoning.
Mapping diagrams
are often used to
represent functions.
Take note of the
pairings to make your
decision.
Solution
a. Input
4
8
2
b. Input Output
Output
1
2
3
The pairing is a function
because each input is
paired with exactly one
output.
2
2
2
4
3
6
4
8
The pairing is not a
function because the
input 2 is paired with
2 and 4.
Checkpoint Tell whether the pairing is a function.
2. Input
Output
5
5
10
15
3
4
6
8
No, the pairing is not a function.
3. Input
Output
0
4
12
20
3
5
9
13
Yes, the pairing is a function.
Copyright © Holt McDougal. All rights reserved.
Lesson 1.6 • Algebra 1 Notetaking Guide
19
Your Notes
A function may be
represented using a
rule that relates one
variable to another.
FUNCTIONS
Verbal Rule
Equation
Table
The output is
2 less than
the input.
Example 3
Input
2
4
6
8
10
Output
Make a table for a function
The domain of the function y 5 3x is 0, 1, 2, and 3.
Make a table for the function, then identify the range
of the function.
Solution
x
y 5 3x
The range of the function is
Example 4
.
Write a function rule
Write a rule for the function.
Input
3
5
7
9
11
Output
6
10
14
18
22
Solution
Let x be the input and let y be the output. Notice that
each output is
the corresponding input. So, a rule
for the function is
.
Checkpoint Write a rule for the function. Identify the
Homework
domain and the range.
4. Yarn (yd)
Total Cost ($)
20 Lesson 1.6 • Algebra 1 Notetaking Guide
1
2
3
4
1.5
3
4.5
6
Copyright © Holt McDougal. All rights reserved.
Your Notes
A function may be
represented using a
rule that relates one
variable to another.
FUNCTIONS
Verbal Rule
Equation
y5x22
The output is
2 less than
the input.
Example 3
Table
Input
2
4
6
8
10
Output
0
2
4
6
8
Make a table for a function
The domain of the function y 5 3x is 0, 1, 2, and 3.
Make a table for the function, then identify the range
of the function.
Solution
0
1
2
3
3(0) 5 0
3(1) 5 3
3(2) 5 6
3(3) 5 9
x
y 5 3x
The range of the function is 0, 3, 6, and 9 .
Example 4
Write a function rule
Write a rule for the function.
Input
3
5
7
9
11
Output
6
10
14
18
22
Solution
Let x be the input and let y be the output. Notice that
each output is twice the corresponding input. So, a rule
for the function is y 5 2x .
Checkpoint Write a rule for the function. Identify the
Homework
domain and the range.
4. Yarn (yd)
Total Cost ($)
1
2
3
4
1.5
3
4.5
6
Function Rule: y 5 1.5x
Domain: 1, 2, 3, 4
20 Lesson 1.6 • Algebra 1 Notetaking Guide
Range: 1.5, 3, 4.5, 6
Copyright © Holt McDougal. All rights reserved.
1.7
Represent Functions as Graphs
Goal p Represent functions as graphs.
Your Notes
GRAPHING A FUNCTION
• You can use a graph to represent a
.
• In a given table, each corresponding pair of input and
output values forms an
.
• An ordered pair of numbers can be plotted as a
.
• The x-coordinate is the
.
• The y-coordinate is the
.
• The horizontal axis of the graph is labeled with the
.
• The vertical axis is labeled with the the
.
Graph a function
Example 1
Graph the function y 5 x 1 1 with domain 1, 2, 3, 4,
and 5.
Solution
Step 1 Make an
table.
x
y
Step 2 Plot a point for each
7
(x, y).
y
6
5
4
3
2
1
O
Copyright © Holt McDougal. All rights reserved.
1
2
3
4
5
6
7
8 x
Lesson 1.7 • Algebra 1 Notetaking Guide
21
1.7
Represent Functions as Graphs
Goal p Represent functions as graphs.
Your Notes
GRAPHING A FUNCTION
• You can use a graph to represent a function .
• In a given table, each corresponding pair of input and
output values forms an ordered pair .
• An ordered pair of numbers can be plotted as a
point .
• The x-coordinate is the input .
• The y-coordinate is the output .
• The horizontal axis of the graph is labeled with the
input variable .
• The vertical axis is labeled with the the output
variable .
Graph a function
Example 1
Graph the function y 5 x 1 1 with domain 1, 2, 3, 4,
and 5.
Solution
Step 1 Make an input-output table.
x
1
2
3
4
5
y
2
3
4
5
6
Step 2 Plot a point for each ordered pair (x, y).
7
y
6
5
4
3
2
1
O
Copyright © Holt McDougal. All rights reserved.
1
2
3
4
5
6
7
8 x
Lesson 1.7 • Algebra 1 Notetaking Guide
21
Your Notes
Example 2
Write a function rule for a graph
Write a function rule for the function represented by the
graph. Identify the domain and the range of the funtion.
7
y
6
5
4
3
2
1
O
2
4
6
8 10 12 14 16 x
Solution
Step 1 Make a
for the graph.
x
y
Step 2 Find a
output values.
Step 3 Write a
relationship.
between the input and
that describes the
y5
A rule for the function is y 5
. The
domain of the function is
The range is
22 Lesson 1.7 • Algebra 1 Notetaking Guide
.
.
Copyright © Holt McDougal. All rights reserved.
Your Notes
Example 2
Write a function rule for a graph
Write a function rule for the function represented by the
graph. Identify the domain and the range of the funtion.
7
y
6
5
4
3
2
1
O
2
4
6
8 10 12 14 16 x
Solution
Step 1 Make a table for the graph.
x
2
4
6
8
10
y
0
1
2
3
4
Step 2 Find a relationship between the input and
output values.
From the table, each output value is 1 less
than half the corresponding input value.
Step 3 Write a function rule that describes the
relationship.
1
y 5 }x 2 1
2
1
A rule for the function is y 5 } x 2 1 . The
2
domain of the function is 2, 4, 6, 8, and 10 .
The range is 0, 1, 2, 3, and 4 .
22 Lesson 1.7 • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Checkpoint Complete the following exercise.
1
1. Graph the function y 5 }
x 1 1 with domain 0, 3, 6,
3
9, and 12.
7
y
6
5
4
3
2
1
O
2
4
6
8 10 12 14 x
Checkpoint Write a rule for the function represented
by the graph. Identify the domain and the range of the
function.
2.
14
22
3.
y
14
12
12
10
10
8
8
6
6
4
4
2
2
O
1 2
3
4
5
6 x
24
O
y
2 4
6
8 10 12 x
Homework
Copyright © Holt McDougal. All rights reserved.
Lesson 1.7 • Algebra 1 Notetaking Guide
23
Your Notes
Checkpoint Complete the following exercise.
1
1. Graph the function y 5 }
x 1 1 with domain 0, 3, 6,
3
9, and 12.
x
0
3
6
9
12
y
1
2
3
4
5
7
y
6
5
4
3
2
1
O
2
4
6
8 10 12 14 x
Checkpoint Write a rule for the function represented
by the graph. Identify the domain and the range of the
function.
2.
14
22
Homework
3.
y
14
12
12
10
10
8
8
6
6
4
4
2
2
O
1 2
3
4
5
6 x
24
O
y
2 4
6
8 10 12 x
y 5 3x 2 3
1
y5}
x15
Domain: 1, 2, 3, 4
Domain: 2, 4, 6, 8
Range: 0, 3, 6, 9
Range: 6, 7, 8, 9
Copyright © Holt McDougal. All rights reserved.
2
Lesson 1.7 • Algebra 1 Notetaking Guide
23
Focus On
Functions
Use after Lesson 1.7
Your Notes
Determine Whether a Relation
Is a Function
Goal p Determine whether a relation is a function when
the relation is represented by a table or a graph.
VOCABULARY
Relation
Determine whether a relation is a function
Example 1
Determine whether the relation is a function. Explain
your reasoning.
a.
b.
Input
3
4
2
0
Input
1
2
1
3
4
Output
4
4
5
5
Output
4
7
5
0
3
Solution
a. Every
relation
has one
a function.
b. The input
So,
has
. The
different
.
.
Checkpoint Determine whether the relation is a
function.
1.
Input
4
3
5
4
Output
6
5
7
2
Input
1
2
5
7
Output
4
5
7
6
2.
24 1.7 Focus On Functions • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Focus On
Functions
Use after Lesson 1.7
Your Notes
Determine Whether a Relation
Is a Function
Goal p Determine whether a relation is a function when
the relation is represented by a table or a graph.
VOCABULARY
Relation Any pairing of a set of inputs with a set
of outputs.
Determine whether a relation is a function
Example 1
Determine whether the relation is a function. Explain
your reasoning.
a.
b.
Input
3
4
2
0
Input
1
2
1
3
4
Output
4
4
5
5
Output
4
7
5
0
3
Solution
a. Every input
relation
is
has one output . The
a function.
b. The input 1 has two different outputs .
So, the relation is not a function .
Checkpoint Determine whether the relation is a
function.
1.
Input
4
3
5
4
Output
6
5
7
2
The relation is not a function.
2.
Input
1
2
5
7
Output
4
5
7
6
The relation is a function.
24 1.7 Focus On Functions • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
VERTICAL LINE TEST
A graph of a relation is a
passes through more than one
y
if no vertical line
on the graph.
y
x
y
x
x
Use the vertical line test
Example 2
Determine whether the graph represents a function.
a.
b.
y
y
4
4
2
2
O
x
x
2
4
O
6
A vertical line
be drawn through more
than one point. The graph
a
function.
2
6
4
A vertical line
be drawn through
more than one point.
The graph
a function.
Checkpoint Determine whether the graph represents a
function.
Homework
3.
4.
y
4
y
4
2
2
x
O
Copyright © Holt McDougal. All rights reserved.
2
4
6
x
O
2
4
6
1.7 Focus On Functions • Algebra 1 Notetaking Guide
25
Your Notes
VERTICAL LINE TEST
A graph of a relation is a function if no vertical line
passes through more than one point on the graph.
y
y
x
y
x
Function
x
Not a function
Not a function
Use the vertical line test
Example 2
Determine whether the graph represents a function.
a.
b.
y
y
4
4
2
2
O
x
x
2
4
O
6
A vertical line cannot
be drawn through more
than one point. The graph
represents
a
function.
2
6
4
A vertical line
can
be drawn through
more than one point.
The graph does not
represent a function.
Checkpoint Determine whether the graph represents a
function.
Homework
3.
4.
y
4
y
4
2
2
x
O
2
4
6
function
Copyright © Holt McDougal. All rights reserved.
x
O
2
4
6
not a function
1.7 Focus On Functions • Algebra 1 Notetaking Guide
25
Words to Review
Give an example of the vocabulary word.
Variable
Algebraic expression
Power, Base, Exponent
Verbal model
Rate
Unit rate
Equation
Inequality
Formula
Function
Domain
Range
Relation
Review your notes and Chapter 1 by using the
Chapter Review on pages 54–57 of your textbook.
26 Words to Review • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Words to Review
Give an example of the vocabulary word.
Variable
Algebraic expression
x, where x is any real
number
3x
Power, Base, Exponent
Verbal model
23
2 is the base, 3 is the
exponent
number of CDs 4
number of customers
Rate
Unit rate
}
80 miles
4 hours
20 mi/h
Equation
Inequality
2a 1 4 5 10
5n ≤ 20
Formula
Function
d 5 rt
y 5 5x
Domain
Range
Input
0
1
2
Input
0
1
2
Output
2
4
6
Output
2
4
6
The domain is 0, 1, 2 for
this input-output table.
The range is 2, 4, 6 for
this input-output table.
Relation
Input
1
1
2
Output
2
3
4
Review your notes and Chapter 1 by using the
Chapter Review on pages 54–57 of your textbook.
26 Words to Review • Algebra 1 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.