1. Copy entire table into notebook+ - x ÷ =
( )
< >
increased by
decreased by of
quotient is
quantity
is greater than
more than
minus
product to
are
sum
is less than
add
subtract
times
divided among
was
difference
is no more than
total
less than
were
is at least/most

2. Variables and Expressions

3. Objectives Vocabulary variable Constant algebraic expression evaluate
Translate between words and algebra.
Evaluate algebraic expressions.
Vocabulary
variable
Constant
algebraic expression
evaluate

4. A variable is a letter or a symbol used to represent a value that can change.
A constant is a value that does not change.
An algebraic expression may contain variables, constants, and operations.
Two numbers are opposites if their sum is 0.
Additive inverses are a number and its opposite.
Two numbers are reciprocals if their product is 1.
A number and its reciprocal are called multiplicative inverses. To divide by a number, you can multiply by its multiplicative inverse.

5.   + – Plus, sum, increased by Minus, difference, less than
You will need to translate between algebraic expressions and words to be successful in math. + –  
Plus, sum,
increased by
Minus, difference, less than
Times, product, equal groups of
Divided by, quotient

6. These expressions all mean “2 times y”:
2y (y)
2•y (2)(y)
2 * y (2)y
Writing Math

7. Translating from Algebra to Words
Give two ways to write each algebra expression in words.
A r B. q – 3
the sum of 9 and r
the difference of q and 3
9 increased by r
3 less than q
C. 7m D. j ÷ 6
the product of m and 7
the quotient of j and 6
m times 7
j divided by 6

8. Multiply Divide Add Subtract Put together, combine
To translate words into algebraic expressions, look for words that indicate the action that is taking place.
Add
Subtract
Multiply
Divide
Put together, combine
Find how much more or less
Put together equal groups
Separate into equal groups

9. How Do You Describe a Variable Expression?
Meaning
Operation

10. More or Less?
Copy
What to do
Algebraic
Expression
Numerical
Less
Subtract in order
-13 less 5
-13 – 5
Less than
Reverse the order
-13 less than 5
5 – (-13)
Is less than
inequality
-13 is less than 5
-13 < 5
This also works for more, more than, and is more than.

11. Translating from Words to Algebra
John types 62 words per minute. Write an expression for the number of words he types in m minutes.
m represents the number of minutes that John types.
62 · m or 62m
Think: m groups of 62 words

12. Translating from Words to Algebra
Roberto is 4 years older than Emily, who is y years old. Write an expression for Roberto’s age
y represents Emily’s age.
y + 4
Think: “older than” means “greater than.”

13. Translating from Words to Algebra
Joey earns $5 for each car he washes. Write an expression for the number of cars Joey must wash to earn d dollars.
d represents the total amount that Joey will earn.
Think: How many groups of $5 are in d?

14. To evaluate an expression is to find its value.
To evaluate an algebraic expression, substitute
numbers for the variables in the expression and
then simplify the expression.

15. Evaluating Algebraic Expression
Evaluate each expression for a = 4, b =7, and
c = 2.
A. b – c
b – c = 7 – 2
Substitute 7 for b and 2 for c.
= 5
Simplify.
B. ac
ac = 4 ·2
Substitute 4 for a and 2 for c.
= 8
Simplify.

16. Try This!
Evaluate each expression for m = 3, n = 2, and
p = 9.
a. mn
mn = 3 · 2
Substitute 3 for m and 2 for n.
= 6
Simplify.
b. p – n
p – n = 9 – 2
Substitute 9 for p and 2 for n.
= 7
Simplify.
c. p ÷ m
p ÷ m = 9 ÷ 3
Substitute 9 for p and 3 for m.
= 3
Simplify.

17. A replacement set is a set of numbers that can
be substituted for a variable. The replacement
set in Example 4 is (20, 50, and 325).
Writing Math

18. Warm Up Simplify. 1. 42 2. |5 – 16| 3. –23 4. |3 – 7| 8  6 10y – 4 1
|5 – 16| – |3 – 7| 16 11 –8 4
Translate each word phrase into a numerical or algebraic expression.
5. the product of 8 and 6
8  6
6. the difference of 10y and 4
10y – 4
Simplify each fraction. 1 7 16 2 8 56 7. 8 8.
Add.
462
1.80
Multiply.
(22)
28.6
11. 25(8)
200

19. Simplifying expression
Order of Operations; Distributive Property; Combining Like Terms

20. Objective
Use the order of operations to simplify expressions.
Use the Commutative, Associative, and Distributive Properties to simplify expressions.
Combine like terms.
Vocabulary
order of operations
term
like terms
coefficient

21. When a numerical or algebraic expression contains more than one operation symbol, the order of operations tells which operation to perform first.
Order of Operations
Perform operations inside grouping
symbols.
First:
Second:
Evaluate powers.
Third:
Perform multiplication and division
from left to right.
Perform addition and subtraction from
left to right.
Fourth:

22. Order of Operations
Simplify each expression.
A. 15 – 2 · 3 + 1
15 – 2 · 3 + 1
There are no grouping symbols.
15 – 6 + 1
Multiply. 10
Subtract and add from left to right.
B. 12 – ÷ 2
12 – ÷ 2
There are no grouping symbols.
12 – ÷ 2
Evaluate powers. The exponent applies only to the 3.
12 – 9 + 5
Divide.
Subtract and add from left to
right. 8

23. Try This!
Simplify the expression.
–20 ÷ [–2(4 + 1)]
There are two sets of grouping
symbols.
–20 ÷ [–2(4 + 1)]
Perform the operations in the
innermost set.
–20 ÷ [–2(5)]
Perform the operation inside
the brackets.
–20 ÷ –10 2
Divide.

24. Evaluating Algebraic Expressions
Evaluate the expression for the given value of x.
42(x + 3) for x = –2
42(x + 3)
First substitute –2 for x.
42(–2 + 3)
Perform the operation inside the parentheses.
42(1)
16(1)
Evaluate powers. 16
Multiply.

25. Try This!
Evaluate the expression for the given value of x.
(x · 22) ÷ (2 + 6) for x = 6
(x · 22) ÷ (2 + 6)
(6 · 22) ÷ (2 + 6)
First substitute 6 for x.
(6 · 4) ÷ (2 + 6)
Square two.
(24) ÷ (8)
Perform the operations inside the parentheses. 3
Divide.

26. Fraction bars, radical symbols, and absolute-value symbols can also be used as grouping symbols. Remember that a fraction bar indicates division.

27. Simplifying Expressions with Other
Grouping Symbols
Copy
Simplify.
2(–4) + 22
42 – 9
The fraction bar acts as a grouping symbol. Simplify the numerator and the denominator before dividing.
2(–4) + 22
42 – 9
–8 + 22
42 – 9
Multiply to simplify the numerator.
–8 + 22
16 – 9
Evaluate the power in the denominator.
Add to simplify the numerator. Subtract to simplify the denominator. 14 7 2
Divide.

28. Evaluate the power in the denominator.
Try this!
Simplify.
5 + 2(–8)
(–2) – 3 3
The fraction bar acts as a grouping symbol. Simplify the numerator and the denominator before dividing.
5 + 2(–8)
(–2) – 3 3
5 + 2(–8)
–8 – 3
Evaluate the power in the denominator.
5 + (–16)
– 8 – 3
Multiply to simplify the numerator.
–11
Add. 1
Divide.

29. You may need grouping symbols when translating from words to numerical expressions.
Remember!
Look for words that imply mathematical operations.
difference subtract
sum add
product multiply
quotient divide

30. Translating from Words to Math
Translate each word phrase into a numerical or algebraic expression.
A. the sum of the quotient of 12 and –3 and the square root of 25
Show the quotient being added to
B. the difference of y and the product of 4
and
Use parentheses so that the product is evaluated first.

31. Try This!
Translate the word phrase into a numerical or algebraic expression: the product of 6.2 and the sum of 9.4 and 8.
Use parentheses to show that the sum of 9.4 and 8 is evaluated first.
6.2( )

32. Try This!
Translate the word phrase into a numerical or algebraic expression:
3 three times the sum of –5 and n
Use parentheses to show that the sum of -5 and n is evaluated first.
3(-5 + n)

33. The Commutative and Associative Properties of Addition and Multiplication allow you to rearrange an expression to simplify it.
Copy

34. Try This!
Simplify.
Use the Commutative Property.
Use the Associative Property to make groups of compatible numbers.
( ) + (58 + 2)
(500) + (60)
560

35. ( ) Try This! Simplify. 1 2 • 7 • 8 1 2 8 • 7
Use the Commutative Property. ( ) 1 2 • 8 7
Use the Associative Property to make groups of compatible numbers. 4 • 7 28

36. The Distributive Property is used with Addition to Simplify Expressions.
Copy
The Distributive Property also works with subtraction because subtraction is the same as adding the opposite.

37. Using the Distributive Property with Mental Math
Write the product using the Distributive Property. Then simplify.
5(59)
5(50 + 9)
Rewrite 59 as
5(50) + 5(9)
Use the Distributive Property.
Multiply.
295
Add.

38. The terms of an expression are the parts to be added or subtracted
The terms of an expression are the parts to be added or subtracted. Like terms are terms that contain the same variables raised to the same powers. Constants are also like terms.
Like terms
Constant
Copy
4x – 3x + 2
A coefficient is a number multiplied by a variable. Like terms can have different coefficients. A variable written without a coefficient has a coefficient of 1.
Coefficients
Copy
1x2 + 3x

39. Copy
Using the Distributive Property can help you combine like terms. You can factor out the common factors to simplify the expression.
7x2 – 4x2 = (7 – 4)x2
Factor out x2 from both terms.
= (3)x2
Perform operations in parenthesis.
= 3x2
Notice that you can combine like terms by adding or subtracting the coefficients and keeping the variables and exponents the same.

40. Copy Combining Like Terms
Simplify the expression by combining like terms.
Copy
72p – 25p
72p – 25p
72p and 25p are like terms.
47p
Subtract the coefficients.

41. Copy Combining Like Terms
Simplify the expression by combining like terms.
Copy
0.5m + 2.5n
0.5m + 2.5n
0.5m and 2.5n are not like terms.
0.5m + 2.5n
Do not combine the terms.

42. Try This!
Simplify by combining like terms.
3a. 16p + 84p
16p + 84p
16p + 84p are like terms.
100p
Add the coefficients.
3b. –20t – 8.5t2
–20t – 8.5t2
20t and 8.5t2 are not like terms.
–20t – 8.5t2
Do not combine the terms.
3c. 3m2 + m3
3m2 + m3
3m2 and m3 are not like terms.
3m2 + m3
Do not combine the terms.

43. Simplifying Algebraic Expressions
Copy
Simplifying Algebraic Expressions
Simplify 14x + 4(2 + x). Justify each step.
Procedure
Justification 1.
14x + 4(2 + x) 2.
14x + 4(2) + 4(x)
Distributive Property 3.
14x x
Multiply.
Commutative Property 4.
14x + 4x + 8 5.
(14x + 4x) + 8
Associative Property 6.
18x + 8
Combine like terms.

44. Try This!
Simplify 6(x – 4) + 9. Justify each step.
Procedure
Justification 1.
6(x – 4) + 9 2.
6(x) – 6(4) + 9
Distributive Property 3.
6x –
Multiply.
Combine like terms. 4.
6x – 15

45. Lesson Quiz: Part I
Give two ways to write each algebraic
expression in words.
1. j – 3
2. 4p
3. Mark is 5 years older than Juan, who is y years old. Write an expression for Mark’s age.
The difference of j and 3; 3 less than j.
4 times p; The product of 4 and p.
y + 5

46. Lesson Quiz: Part II
Evaluate each expression for c = 6, d = 5, and e = 10.
c + d
Shemika practices basketball for 2 hours each
day. d e 1 2 11
6. Write an expression for the number of hours she
practices in d days.
7. Find the number of hours she practices in 5, 12,
and 20 days. 2d
10 hours; 24 hours; 40 hours

47. 4. 3 three times the sum of –5 and n 3(–5 + n)
Lesson Quiz: Part III
Simply each expression. 2.
52 – (5 + 4)
|4 – 8|
1. 2[5 ÷ (–6 – 4)] –1 4
3. 5  8 – ÷ 22 40
Translate each word phrase into a numerical or algebraic expression.
4. 3 three times the sum of –5 and n
3(–5 + n)
5. the quotient of the difference of 34 and 9 and the square root of 25
Simplify each expression by combining like terms. Justify each step with an operation or property.
6. 14c2 – 9c
14c2 – 9c
7. 301x – x
300x
8. 24a + b2 + 3a + 2b2
27a + 3b2