Algebra
Expressions
and
Equations
Name ______________________
Color __________
Properties of Real Numbers
Property
Example
Commutative Property of Addition
5+6 =6+5
11 = 11
Associative Property of Addition
3 + (4 + 5) = (3 + 4) + 5
3+ 9 = 7
+5
12
=
12
Identity for Addition
8+0=8
0 is the Identity of Addition
Additive Inverse
8 + (–8) = 0
What you add to get zero!
Commutative Property of Multiplication
7x8=8x7
56 = 56
Associative Property of Multiplication
5 x (3 x 4) = (5 x 3) x 4
5 x 12 = 15 x 4
Identity for Multiplication
4x1=4
1 is the Identity for Multiplication
Multiplicative Inverse
4x¼=1
What you multiply to get 1!
Distributive Property
4(3 + 7) = 4(3) + 4(7)
4(10 ) = 12 + 28
40
=
40
Distributive Property
For Algebraic Expression
4(x + 8) = 4x + 32
Use the table above to create your own definition and make your own examples!
Property
Hint (Key word)
Addition
Multiplication
Commutative
Associative
Identity
Inverse
Distributive
2
Properties of Real Numbers
Write the property next to the equation. Use the word bank.
1)
2(–5 + 3) = 2(–5) + 2(3)
____________________________________
2)
11 + (–2) = (–2) + 11
____________________________________
3)
11 + 0 = 11
____________________________________
4)
1
3
•3=1
____________________________________
5)
4 • (–2) = –2 • 4
____________________________________
6)
–7 + ( 1 + 8) = (–7 + 1) + 8
____________________________________
7)
3•0=0
____________________________________
8)
–15 • 1 = –15
____________________________________
9)
–5 • (7 • 4) = (–5 • 7) • 4
____________________________________
10)
9 + (–9) = 0
____________________________________
Word Bank:
Multiplicative Inverse
Commutative (Addition)
Additive Identity
Additive Inverse
Associative (Addition)
Multiplicative Identity
Commutative (Multiplication)
Associative (Multiplication)
Distributive
Multiplicative Property of Zero
3
Properties HW
Application: Apply the listed property to complete the following statements.
1) Additive Identity: 4 + _________ = ______________
2) Associative Property of Addition: (9 + 3) + 5 = ______________________
3) Multiplicative Inverse: 8  ____________ = ____________________
4) Additive Inverse: (–7) + _____________ = _____________________
5) Distributive Property: 4(x + 3) = __________________________
6) Multiplicative Identity: (–14)  ________________ = ____________________
7) Zero Property of Multiplication: 321  _________________ = _________________
*8) Commutative Property of Multiplication: (5 + 6)  3 = __________________________
Practice Questions
1) Mrs. Hansen asked Eli to apply the distributive property to the expression, 2(7 + 3). Which
of the following should Eli have written?
A.
2(10)
B.
2(7) + 2(3)
C.
D. (7 + 3)  2
2(3 + 7)
2) Which is an example of the associative property of multiplication?
A.
7x0x9=0
C.
(6 
B. 4  (7 + 3) = 4  (3+7)
1
)  3=3
6
D. 5  (3  8) = (5  3)  8
3) If a + b = a, then b equals _____.
A –1
B 0
C 1
D –a
4) Circle each of the following that is an example of the Commutative Property of Addition:
7(4 + 2) = 7(4) + 7(2)
–9 + 7 = 7 + (–9)
16  3 = 3  16
7–2=2–7
(3 + 5) + 8 = (5 + 3) + 8
15 + 81 = 81 + 15
7+0=7
(4 + 5)  3 = (5 + 4)  3
4
LIKE TERMS
Yes or No? If Yes, write the answer!

3x + 7x
___________

3ab – 6b
___________

5x + 5y
___________

2a – 5a
___________

4c + c
___________

x + x²
___________

4d+ 4
___________

6 + 10
___________
 Coefficients: a number written in front of the variable.
Example: 6x The coefficient is _____________.
Example: x The coefficient is _____________.
Simplify the following expressions by adding the coefficients:
1.
2x + 4x
2.
3a + 7a
3.
6xy – 2xy
4.
2a + 5a + 6
5.
3½y + 5y – 4y
6.
5d – 6d – 3d
7.
3xy – xy + 2x
8.
cd + 4cd – 2a
9.
4s – 4s
10.
–4c + 8c – 6c
11.
½e – 2e + ¾e
12.
5x + 4x + 4x + 11x
13.
Find the sum of –3x and 8x
14.
Find the sum of –7g and 4g + 2
5
Challenge Questions
1.
–5x – 3x
2.
8x – 2x
3.
–7x – (–3x)
4.
6x – (–4x)
5.
–10x –14x
6.
–9x – (–x)
7.
3x – 8x
8.
x – (–5x)
9.
3(2x)
10.
4y(5) + 7
11.
3(2x) + 4(5y) – 3x
12.
5(6x) – 4(5x) – 10
Word Problems
1.
The angle measures of a triangle are (x–7), (x), and (3x +2). Write an expression in simplest form
to represent the measures of each of the angles of the triangle.
2.
Find the perimeter of the rectangle.
Combine like terms.
A
B
C
D
3.
4x + 3y
8x + 6y
12xy
4x2 + 3y2
Combine Like Terms:
4.8 + 2.2w – 1.4w + 2.4
6
Like Terms & Exponents
You can only add like terms with the SAME variable AND exponents!!!
Like Terms ––> x2 + x2 = 2x2
NOT Like Terms ––> a2 + a3
1.
a2 + b2 +2a2 +5b2
2.
7h2 + 3 – 2h2 +4
3.
3x2 + 3y + x + y + z
4.
5b + 5b + 6b2 – 10 – 3b
5.
Jack got the expression 7x + 1, then wrote his answer as 1 + 7x. Is his answer an equivalent
expression? How do you know?
6.
Jill also got the expression 7x + 1, then wrote his answer as 1x +7. Is her expression an
equivalent expression? How do you know?
7.
Find the sum of 2x + 1 and 5x
9.
Write an expression for the perimeter of the figure to the right.
10.
Write an expression for the perimeter of the rectangle below.
8.
Find the sum of (–3a + 2) and (–5a – 3)
7
Substitution
http://www.mathsisfun.com/algebra/substitution.html
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Practice
Evaluate each expression if d = 8, e = 3, f = 4 and g = –1
1.
2(d + 9)
2.
𝑑
4
3.
𝑒𝑓
4
4.
4f + d
5.
5𝑑−25
5
6.
d2 + 7
7.
𝑑−4
2
8.
10(e + 7)
9.
2𝑔
2
8
10.
8(5m) + 2(3m); m = –2;
11.
–6(2v) + 3a(3); v = 1/3; a = 2/3
Evaluate each expression below. (Turn ÷ into a fraction, substitute, & cancel)
1.
8x ÷ 2; x = –¼
2.
18w ÷ 6; w = 6
3.
25r ÷ 5r; r = –2
4.
33y ÷ 11y; y = –2
5.
56k ÷ 2k; k = 3
6.
24xy ÷ 6y; x = –2; y = 3
Check whether the given value is a solution to the equation
7.
4n – 3 = –2n + 9 for n = 2
8.
9m – 19 = 3m + 1 for m =
9.
3(y + 8) = 2y – 6 for y = 30
10.
–3(8x) = 6(4x) for x = 2
10
3
9
Adding Polynomials
1.
(2x + 3y) + (4x + 9y)
2.
(3x – 5) + (x – 7) + (7x + 12)
3.
–4x3 + 6x2 – 8x – 10 and 7x3 – 4x2 + 9x + 3
4.
(3a + 5b + 7c) + (8a – 2b – 9c)
5.
(7e² + 3e +2) + (9 – 6e + 4e²) + (9e + 2 – 6e²)
6.
(5m – 4q + 9) + (2q + 7)
7.
(a² + a + 3) + (15a² + 2a + 9)
8.
(5x + 2x²) + (3x – 2x²)
10.
Add: 4x² + 6x + 9 to 2x² + 3x – 9
9. (s² + 3s – 3) + (2s² + 9s – 2) + (s – s²)
11.
What polynomial can be added to (2x2 + 3x + 1) to get (2x2 + 8x)?
12.
Find the result when – 18m – 4 is added to 4m – 14
13.
Add: 4x², 6x² – 20, and (–40)
14.
(–x + 10) + (–3x + 6)
10
15.
The side lengths of a triangle can be expressed using the following binomials: (2x + 4), (3x + 5),
and (4x – 7). Write an expression for the perimeter.
16.
Find the sum of (–4x + 5) and (15x – 3)
Common Core Question
Subtract Polynomials
1. (–3x – 2) – (7x + 9)
2.
(–2x – 1) – (x – 7)
3. (9x + 5) – (6x – 8)
4.
(–8x + 1) – (8x – 1)
5. (3p2 – 2p + 3) – (p2 – 7p + 7)
6.
(3a + 5b + 7c) – (5a – 2b + 9c)
7. Subtract: 2x – 4 from –3x + 1
8.
(– 3x2 + 4x – 11) – (–6x2 – 8x + 10)
11
9.
(3x² + 2y2 + 7) – (4x² – 2y2 – 8)
11.
Find the result when – 2m – 3 is added to 2m + 4
12.
Find the result when – 3x + 4 is taken away from –2x + 9
13.
Subtract (5m – 6n + 12) from (2m + 3n – 5).
14.
Subtract 8a + 5b – 6c from 10a + 8b + 7c
16.
Find the difference when 6h is subtracted from 2h – 4.
17.
Find the result when 13v + 2 is subtracted from 11 + 5v
18.
Find the result when –3n – 7 is subtracted from n + 4.
19.
Subtract: 3x – 5 from –8x + 18
10.
(x2 + y2) – (–x2 + y2)
15. (4x + 8y + 9z – 7a + 5b) – (4b + 5x + 7y + 3z + 2a)
20.
Subtract: (x² + 2x) from (12x2 + 5)
12
21. Missy made a frame. She cut a rectangle with an area of (x2 + 3x) square inches from a piece of
wood that had an area of (2x2 + 9x + 10). Write an expression for the area of the remaining frame.
22. Find the result when – 2x + 9 is taken away from –7x + 2
Common Core Question
Write an expression equivalent
The Distributive Property
Simplify the following:

3(x + 6)
______________

–6(3y – 5)
______________

4(4 – y)
______________

3 – 4(x + 6)
______________

–7(2 + z)
______________

2x + 3(5x –3) +5 _____________

(2a + 3)5
______________
 (6a – 7)9
_____________
13
Factoring
Factoring is dividing out the GCF and “un–doing” distribution!
Factor each expression
1. 12a – 6h
2. 3x + 9
3. 12x + y
4. 24a – 4
5. 72a + 9n
6. 8a – 8v
Distribute & Factor Practice
Simplify the following expressions.
1. 2(4 + 9x)
2. 7(x + –1)
3. 12(a + b + c)
4. 7(a + c + b)
5. –10(3 + 2 + 7x)
6. –2(3w + 3x + –2z)
7. –1(x + 2)
8. 3(–2 + 2x2y3 + 3y2)
9. 5(5 + 5x)
10.
y(1 + x)
11.
12x(3x + 3)
12.
9(9x + 9y)
13.
3(m + n)
14.
2 – 9(x –12)
15.
3(–m + 9)
16.
–6(3x – 8)
17.
2(x – y + 1)
18.
4(6p + 2q – 2p)
14
19.
3(2x + 5) – 8
20.
4(– 3x + 6) – 2x
21.
2(9y + 11) + 7
22.
–1(x – 9) + 4x
23.
2+ 4(3x + 7)
24.
3 – 5(7y – 11)
25. 4a – 32
26.
28a + 4t
27.
30a – 6s
28. 7a – 28d
29.
3a + 12h
30.
4a + 32y
31. 9a + 3
32.
9a – 81h
33.
72a + 9z
34. 20a + 4t
35.
6a + 3
36.
9a – 63
Factor each expression
Common Core Exam Question
Distribute & Factor HW#1
15
Simplify (distribute).
1.
¼ (4x + 8)
2.
1
/6 (r – 6)
3.
4
4.
1
5.
¾ (5x – 1)
6.
1
7.
3(2x – 1)
8.
10(b + 4c)
9.
9(g – 5h)
10.
7(4n – 5m – 2)
11.
a(b + c + 1)
12.
(8j – 3w + 9)6
/8(2x + 4)
/5(x + 1)
/5(10x – 5) – 3
Find the GCF of each pair of monomials
1.
7, 28n
2.
30x, 24xy
3. 42mn, 14mn
Factor the following expressions.
1.
81w + 48
2.
10 – 25t
3.
12a + 16b + 8
4.
72t + 8
5.
36z + 72
6.
3r + 3s
7.
6x + 24
8.
8x2 – 4x
9.
6xy + 10x
3.
3x2 – 4x + x
Combine like terms and then factor your answer.
1.
–12m + 18m – 15
2.
m2 – 4m + 3m + 12m
16
Distribute & Factor HW
1.
4(8m – 7n) + 6(3n – 4m)
2.
9(r – s) + 5(2r – 2s)
3.
12(1 – 3g) + 8(g + f)
4.
6(–5r – 4) – 2(r – 7s – 3)
5.
5 – 7(–4q + 5)
6.
–(2h – 9) – 4h
7.
–3(1 – 8m – 2n)
8.
55a + 11
9.
144q – 15
10.
7a + ab
10.
Which pair of monomials has a GCF of 4a?
a.
c.
11.
16a, 8a
16ab, 12b
b. 18a, 8a
d. 16ab, 12a
Which of the following expressions cannot be factored?
a.
c.
6 + 3x
15x + 10
b.
d.
7x + 3
30x + 40
Write the letter on the line that matches each expression with its equivalent expression
12.
3+1
_____
a.
8 – 4x
13.
4(2 – x)
_____
b.
5x + 5
14.
3x – 2 – x + 6
_____
c.
3(x + 7)
15.
2(x + 2) + (3x + 1)
_____
d.
1+3
16.
3x + 21
_____
e.
2x + 4
17
Solving Equations
1.
8a + 5 = 53
2.
–9.4 + z = –3.6
3.
–7 = c – 6
4.
a – 3.5 = 4.9
5.
x – 2.8 = 9.5
6.
2.25 + b = 1
7.
5
11
 c 
6
12
8.
–8.5 + r = –2.1
9.
7
11
 m
9
6
10.
2(b – 2) + b + 3 = 6.5
11.
–3 = –3(2t – 1)
12.
x – 2(x + 10) = 12
13.
Show work:
18
Solving Equations HW
Solve and check.
1.
3x – 10 = 20 _______________
2.
x
+ 3 = 63 __________________
5
3.
16 = 2x + 6 _______________
4.
–2x – 7 = 35 ___________________
5.
–28 = 3x – 1 _______________
6.
9 = x – 2 ____________________
8
7.
2
x
3
8.
–15 = 5(3q – 10) – 5q _________________
9.
42 = 3(2 – 3h) _________________
10.
–10 = 5(2w – 4) _________________
11.
–5w = –24.5 ______________
12.
6(3m + 5) = 66 _________________
– 4 = 20 ______________
19
Writing Algebraic Expressions
 Let Statement: math sentence used to define a __________________ to represent the
_____________________________.
1.
Laura has twice as much homework as Ann.
Let x =
2.
The Bills won five more games than they lost.
3.
The Tigers had three times as many hits as the Yankees.
4.
The length of a rectangle is 3 cm more than the width.
5.
Eight more than twice a number is 32.
6.
Seven more than three times a number is 25.
7.
Twice a number increased by four is 16.
8.
Six less than three times a number is 21.
9.
Fifteen less than twice a number is 25.
10.
Sixty–six is eleven more than five times a
number.
20
Setting up and Solving Word Problems
LESCA





L – Write your let statement
E – Write your equation
S – Solve
C – Check
A – Write an answer sentence
1.
A cell phone company charges $39 a month plus $.15 per text message sent. If Jan sends 35 text
messages this month, how much does she owe before taxes are added?
2.
A concert charges an admittance fee of $12 and $2 for each snack at the snack bar. If you bring $20
how many snacks can you buy at the concert?
3.
A rental car company ABC charges $25 per day plus $.15 per mile. Rental car company XYZ charges
$18 per day plus $.25 per mile. If you plan to drive 50 miles, who is the cheaper rental company?
4.
The admission to an amusement park is $8. Tickets for rides cost $4 each. Joe needs one ticket for
each ride. Write an equation Joe can use to determine the number of ride tickets, r, he can buy if he
has $40 before he pays the admission fee.
5.
Barry’s mountain bike weighs 6 pounds more than Andy’s. If their bikes weigh 42 pounds altogether,
how much does Barry’s bike weigh?
6.
The sum of two consecutive odd numbers is 156. What are the numbers?
21
Solving Equations (LESCA) HW
Write and solve an equation for each problem.
1.
The perimeter of a rectangle is 30 inches. If its length is three times its width, find the
dimensions.
2.
Trevor and Marissa together have 26 t–shirts to sell. If Marissa had 6 fewer t–shirts than Trevor,
find how many t–shirts Trevor has.
3.
4.
A number is 6 greater than ½ another number. If the sum of the numbers is 21, find the
numbers.
5.
A vending machine had twice as many quarters in it as dollar bills. If the quarters and dollar bills
have a combined value of $96, how many quarters are in the machine?
22
7.
A cell phone company has a basic monthly plan of $40 plus $.45 for any minutes uses over 700.
Before receiving his statement, John saw he was charged a total of $48.10. Write and solve an
equation to determine how many minutes he must have used during the month.
8.
A volleyball coach plans her daily practices to include 10 minutes of stretching, 2/3 of the entire
practice scrimmaging, and the remaining practice working on drills of specific skills. On
Wednesday, the coach planned 100 minutes of stretching and scrimmaging. How long, in hours,
is the entire practice?
9.
The sum of two consecutive even numbers is 54. Find the numbers.
10.
Justin has $7.50 more than Eva and Emma has $12 less than Justin does. How much money does
each person have if they have a total of $63.
11.
Suppose you want to buy your favorite ice cream bar while at an amusement park and it costs
$2.89. If you purchase the ice cream bar and 3 bottles of water, and pay with a $10 bill and
receive no change, then how much did each bottle of water cost?
23
Inequalities
 An inequality is a mathematical sentence that compares two quantities that
_____________________________!!!
Use the following symbols to represent inequalities:
_______ means “is less than.”
_______ means “is less than or equal to.”
_______means “is greater than.”
_______ means “is greater than or equal to.”
_______ means “is not equal to.”
What’s the difference?

______________________ means that x is less than 4.
o

What is a number in this solution set? ______________
______________________ means that x is less than 4 OR equal to 4.
o
What is a number in this solution set? ______________
You graph your inequalities on a number line.

This graph shows the inequality _____________________

A _______________________ means that’s where the graph starts, but 4 is not part of the
graph.

A _______________________ represent all the numbers less than 4.

What is this inequality?
_______________________

What is this inequality?
_______________________
24
Graphing Solutions

Use an _______________ ________________ (
) to graph inequalities with
< or > signs because the number is NOT part of the graph

Use a _________________ _______________ (

or

) to graph inequalities with
signs because the number IS part of the graph.
Graph each inequality:
x < 4 (A number less than 4)
x < 6 (A number less than or equal to 6)
x > –3 (A number greater than –3)
Practice
1. x < 3
2. x > –5
3. x < –1
4. x > 2
25
Solving Inequalities
 Solve an inequality just as you would an equation.
3x – 15 > 12
Write the inequality.
 When the solution of an inequality requires dividing or multiplying by a negative number, you
need to change the direction of the inequality sign. Compare these solutions.
12x < 60
–12x < 60
 Choose a number included in the solution to 12x < 60. Is this also a solution to –12x < 60?
Explain.
 When you divide by a negative coefficient in an inequality you _____________________________
Practice
1.
x+7<9
2.
y – 12 > 3
3.
7x – 49 > 98
4.
5 – 3a < 13
5.
–2b < 10
6
–55 < – 5c
7.
p + 6.8 > 14
8.
36 > –1.8y
26
Inequalities HW
Directions: Solve, Graph, and Check each inequality.
1.
5x – 8 < 17
2.
– 4n + 8 < –4
3.
𝑤
5
+4 > 9
Solve and Check
5.
1
5
1.
4x – 2 < 26
2.
6– 𝑦<7
3.
2x + 27 > 15
4.
10x > 14x + 8
4 + 7x < 25
27
Writing Inequalities Using Symbols
Like equations, you can write inequalities to represent a situation.
 How could you represent, Lisa will spend less than $25?
 How could you represent, Rodney ran at least 30 miles last week?
 How could you use a number line to show greater than 2?
Independent Practice
1.
Caitlyn volunteers with some friends at a community center.
Caitlyn is shopping online to find a new television for the center. Caitlyn wants a television with
at least a 26–in. screen. Write an inequality to show how much money, m, the center will need
to spend.
2.
Caitlyn drew the graph below to represent the solution to her inequality.
a.
Why do you think she drew a solid circle at 330?
b.
Caitlyn wants to have money left over for accessories. How can Caitlyn change her graph
to show that they need to have more than $330? Draw a graph to show how much they
need to have.
28
3.
The center has a stand for the television that will support up to 30 lb of weight. Draw a graph to
show how much the television she buys can weigh.
4.
A machinist making steel rods for an airplane engine knows that each rod must be at least 9 mm
long but no longer than 9.5 mm.
a.
Write two inequalities that together represent the possible lengths. Graph the
inequalities.
b.
Can both solutions be shown on one graph? If so, draw the graph.
5.
Playing a video game, Emily has gained some points, lost 107 points, and finishes at less than
800 points. The inequality p – 107 < 800 represents this situation. Solve and graph the
inequality.
6.
On level 2 of a video game, the maximum number of points is 1,000. Emily has lost 279 points
and is on level 2.The inequality p – 279 < 1,000 represents this situation. Which is the graph of
its solution?
29
Use the table for 7 – 8
7.
Camille has a goal of hiking more than 350 miles this year. She already hiked the Florida Trail
and now plans to hike 9 miles each day for d days. The inequality 9d + 71 > 350 represents this
situation. Solve and graph the inequality.
8.
Multiple Choice Camille’s brother Roberto hiked the Florida Trail with her and the Myakka
Hiking Trail alone. He wants to hike no more than 400 miles this year and now plans to hike d
day trips of 10 miles each. Which inequality could represent this situation?
A.
10d + 120 > 400
B.
10d + 120 > 400
C.
10d + 120 < 400
D.
10d + 120 < 400
Common Core Question
9.
For her cell phone plan, Heather pays $30 per month plus $.05 per text. She wants to keep her
bill under $60 per month. Which inequality represents the number of texts, t, Heather can send
each month while staying within her budget?
10.
Jenna has $39 to spend on materials to make pottery figures. It costs her $4 to make one figure.
Write and solve an inequality to represent this situation.
30