Name______________________________
Unit 2
Solving Equations and
Properties of Real
Numbers
1
2
Notes 2-1: PROPERTIES of REAL NUMBERS
Important Terminology:
Real Numbers
Associative
Identity
Term
Factor
Inverse
Commutative
Substitution
 COMMUTATIVE PROPERTY
for ADDITION:
for MULTIPLICATION:
In other words…
On your own, make an example using Real Numbers:
 ASSOCIATIVE PROPERTY
for ADDITION:
for MULTIPLICATION:
In other words…
On your own, make an example using Real Numbers:
 SUBSTITUTION PROPERTY:
Example:
On your own, make an example using Real Numbers:
3
PROPERTIES for the OPERATION of ADDITION
 IDENTITY PROPERTY of ADDITION:
In other words…
On your own, make an example using Real Numbers:
 PROPERTY of OPPOSITES
(or ADDITIVE INVERSE):
In other words…
On your own, make an example using Real Numbers:
 DEFINITION of SUBTRACTION:
In other words…
On your own, make an example using Real Numbers:
Identify the property illustrated in each example. All variables represent Real numbers.
1.
1234 + 0 = 1234
________________________________________________
2.
(-8) + 8 = 0
________________________________________________
3.
6   x  2   3 y    x  2   3 y   6
________________________________________________
4.
 5 7  3 7   3 7    57 
________________________________________________
5.
If (a + b) +0 = x, then (a + b) = x
________________________________________________
4
6.
9+0=0+9
________________________________________________
7.
 a  8  (8)  a  8  (8) 
________________________________________________
8.
If x  y  2 and y  c , then x  c  2
________________________________________________
9.
ac 8  8ac
________________________________________________
10.
 x  (7)   x  7
________________________________________________
11.
 2a  x   2  ax 
________________________________________________
12.
10 .3 5    10   0.3    5
________________________________________________
13.
If p  5 then 3  p  2
________________________________________________
 LIKE TERMS (or “Similar Terms”): Have the same variables that also have the same exponents.
 x 2 y 13 5 x 14 x 2 y 2 xy 3
1, 234,567 x  12 xy 3
 21x 0 9 x 2 y 101x 5 xy 3
7 x 2 y  3x 2 y  2 x 8  4 xy 3
Simplify the following algebraic expressions:
EX 1: 5x  7 x  8  2x
EX 2:  y  3z  4 y  9 z  y
EX 3: 2m   6m  6n 
EX 4: 8t  8u 10t 15u
EX 5: 2 x  10 y  21x  3 y 
EX 6: 4e  12 f  3e   15 f  21e
EX 7: 14r  28s  2r  3s  6r
5
Practice 2-1: PROPERTIES of REAL NUMBERS
*Do your practice problems on separate paper with plenty of space to show your work. CHECK YOUR
ANSWERS TO THE ODD PROBLEMS USING THE ANSWER KEY PROVIDED!
I. Name the property illustrated:
1.
2b  6  a  2b  6  a 
2.
If x  5 and  5  32b  q , then x  32b  q
3.
4x  7  7  4x
4.
5a + 0 – 7 = 5a - 7
5.
2bx  (2bx )  0
6.
5x  6y  z  0  5x  6y  z
7.
5  q  (3)  5  (3)  q
8.
b + 2 = z and b = -5 then -5 + 2 = z
9.
4a + 3 – 7c = 4a + 3 + (-7c)
II.
Simplify (the directions could also read “combine similar terms”)
1.
4  2x  (5)
2.
6x  4y  9x  5y
3.
7a  6g  5a  13g
4.
5e  13f  16e  2f  4e
5.
6m  8p  6m
6.
12c  8b  4c  7b
7.
12   5b  2a   12  15a
8.
5c  35h  13h  8c  2h  2
9.
20c  12s  16p  ( p  7s  2c )  14c
10.
32p  16t  88p  2p  6  12t
III.
Replace the “?” with a number that makes the statement true. Then name the property
illustrated.
1.
86  6?
2.
40?
6
3.
19  2  50  ? 19  2
4.
5  ?  2  5   6  2
5.
7  34  7  ?
6.
If u  3 then 3u  11  ?
IV.
Perform the following operations making sure the answer is simplified.
1.
25  5   7  3 2
2.
22  7  52
3.
 4  8  2  3  7   16  2  6


4.
15  12  22  3  1  4 


5.
4 5  7 
 6  2  4
6.


2
6  11
8  7
TURN THIS PAGE UPSIDE DOWN TO CHECK YOUR ODD ANSWERS WITH THIS KEY:
7
Notes 2-2: More Properties of Real Numbers
Important Terminology:
Multiplicative Identity
Reciprocal
Multiplicative Inverse
Distribution
 IDENTITY PROPERTY of MULTIPLICATION:
In other words…
On your own, make an example using Real Numbers:
_____________________________________________________________________________
 PROPERTY of RECIPROCALS:
(or MULTIPLICATIVE INVERSE)
In other words…
On your own, make an example using Real Numbers:
_____________________________________________________________________________
 DEFINITION of DIVISION:
In other words…
On your own, make an example using Real Numbers:
_____________________________________________________________________________
 DEFINITION of DIVISION:
In other words…
On your own, make an example using Real Numbers:
8
I. Name the property illustrated.
1.
x+w=w+x
2.
Z+0=Z
3.
5(7  9) = (7  9)5
4.
 29  3 16    12  29  3 16   12 
5.
2
2
1 
3
3
6.
5 r  5 r
7.
2  2 
   0
3  3 
8.
 2  3 
   1
 3  2 
9.
3(f + g) = 3f + 3g
10.
6 + 29 = 6 + 92
11.
(7 + 6) + 3 = 7 + (6 + 3)
12.
 0 
13.
6ab  6ac  6a(b  c)
14.
(15  2)3 = 3(15  2)
15.
u  (u )  0
16.
1
1
1
13  11  13  11
4
4
4
17.
If x  7  z and z  9, then x  7  9
9
II.
Simplify and circle your answer:
1. 6(3n  2)
2. 4(3 x  5 y )
3. (6m  3n)2
6 3
4. 40   
5 4
 5 3 
5. 24   
 6 8
6. 7 x  5 y  x  5 y  (2 x)
7. 4 z  (10 y)  (8 x)  12 y  17 x  13 z
8. 4(r  5)  9(r  2)  3(4  3r )
9. 2( x  2 z )  3( x  3z )  5(2 x  z )
10. 6b 2  2b 4  8b  5b 2  b 4  11b 2
11. 
11
2
p p
15
3
12. 3.6q  9.3q  2.7q
10
Practice 2-2: More Properties of Real Numbers
*Do your practice problems on separate paper with plenty of space to show your work. CHECK YOUR
ANSWERS TO THE ODD PROBLEMS USING THE ANSWER KEY PROVIDED!
I.
1.
Combine similar terms making sure the answer is simplified.
1
2.
2(3  2r )  6r
 6m  2
3
3.
3q  6m  5q  m  q
4.
 4w
6.
5h  8  19  18h  h
7.
3a  4c  a  3b  c  9a
8.
5  a 2  3a   3  4a 2  5   a
9.
6 3y 2  2y  4   4 2y 2  5y 
10.
3 2y  3  y  1   5   3 5  3y 
11.
a 2  b  4a 2  6b  4a 2
12.
6n 4  3n  6n  n 4  5n 4
13.
9 p 2  4 pt  t 2  9 p 2  6pt  10t 2
14.
5  y  3t   2  4y  t 
15.
7  2a  b   11a  2b 
16.
5r   3r 2   4 8  3 2r 2  r 
17.

18.
1  1
 1 
25    a  b    56a  21b 
5  7
 5 
19.

 1 
1

8 m    w    ( 4)  w  2m      w  2
 4 
2


20.
1
1
1 
1
 5k  10h   2k  (8h )  6  k    h 
5
2
 3 
2
4
 2w 2   w  6w 4  5.
1
 24k  40h 
8
4y 4  3y 3  y 4
11
II.
Match each statement with its matching property:
1.
4(a + b + c) = 4a + 4b + 4c
2.
If a = 3 and b = 4, and a + b + c = 24,
a) additive inverse (property of
opposites)
b) additive identity element
then 3 + 4 + c = 24
3.
7 +3 +  -4  = 7 + 3 +  -4 
c) associative for +
4.
 -8  +  -4   +  -2  =  -4  +  -8  +  -2 
d) associative for x
5.
-432 + 0 = -432
e) commutative for +
6.
xy(1) = xy
f) commutative for x
7.
a + (-a) = 0
g) definition of division
8.
(x + y) +z = (y + x) +z
9.
8 – 3 – 9 = 8 +(-3) + (-9)
i) distributive
10.
1
1 = a × 
a
j) identity element for +
11.
m×8 =8×m
k) identity element for x
12.
 -2 
3
18 ÷   = 18 ×  
3
 -2 
l) multiplicative inverse (property of
h) definition of subtraction
reciprocals)
13.
(38 + t) + 46 = (t +38) + 46
m) substitution
TURN THE PAGE UPSIDE DOWN TO CHECK YOUR ODD ANSWERS WITH THIS KEY:
12
Notes 2-3: Solving Equations
On your own, write a possible definition for the word “Solution.”
 Addition Property of Equality:
 Subtraction Property of Equality:
 Multiplication Property of Equality:
 Division Property of Equality:
Steps in solving equations:
1.
2.
3.
4.
5.
For each problem, determine steps listed.
1)
2(5a  1)  4
2)
1
d  20  16
10
Step 1-
Step 1-
Step 2-
Step 2-
Step3-
Step3-
Step 4-
Step 4-
Step 5-
Step 5-
13
3)
3.4  0.5k  0.8
4)
22  5x   16
Step 1-
Step 1-
Step 2-
Step 2-
Step3-
Step3-
Step 4-
Step 4-
Step 5-
Step 5-
5)
45  10  n  3
6)
Step 1-
Step 1-
Step 2-
Step 2-
Step3-
Step3-
Step 4-
Step 4-
Step 5-
Step 5-
13h 19h  8
14
Practice 2-3: Solving Equations
*Do your practice problems on separate paper with plenty of space to show your work. CHECK YOUR
ANSWERS TO THE ODD PROBLEMS USING THE ANSWER KEY PROVIDED!
1.
19  64 15x
2.
83  2x  179
3.
13a   76   119
4.
2
p  10  0
7
5.
0.2  5 12w
6.
2.7h  1.2  1.5
7.
10  63r  17
8.
1
35  32  a
3
9.
k
 11  20
6
10.
6x  0.8  5.2
11.
3.6 y  2.9 y  13
12.
38  9 y   14 y   54
II.
Simplify.
1.
2
5
1



15  8 j  f  5   18  4 f  j  
5
9
2



2.
7  4  2a  3b  5   3  15a    2  3a  5b 
3.
2  m  n      m  n    5  m  2n 
4.
1
2
3
4a 2  3a  6    9  12a 2  3a   16a 2  8a  20 

2
3
4
15
III.
Name the property or definition illustrated.
1.
If k  12 and 12  x  y , then k  x  y
2.
 6  4   8  6   4  8
3.
9+0=0+9
4.
1234 + 0 = 1234
5.
(-8) + 8 = 0
6.
x  y  x
7.
a(b  c)  a b   c  
8.
a c  1    ac  1
IV.
Evaluate given the values for the variables.
1.
5n
n
if n = -3 and p = 4

n p r 6
2.
2 pr  n if n = -3, p = 4, and r = -1
1
y
TURN THE PAGE UPSIDE DOWN TO CHECK YOUR ODD ANSWERS WITH THIS KEY:
16
Notes 2-4: More Equations (  and  )
Use the following problems to review the steps for solving equations. Some have been worked out already as
examples.
4 a  9  3
32  y  11
 9  9
4a  12
EX1:
4a
12

4
4
a  3
EX 2:
3
x  2x  3  6
2 x  5  7 x  15
 5 x  5  15
 5  5
EX 3:
 5 x  10
EX 4:
5 x 10

5 5
x  2
2
2b  5  b  1  3  16
32  2n  3n  5n
 2b  5b  5  3  16
 7b  2  16
 2  2
EX 5:
EX 6:
 7b  14
7b 14

7 7
b  2
2
17
4  k  7   15  3   13
3  6r  5   9  2r  3   12
 18r  15  18r  27  12
12  12
EX 7:
EX 8:
This is a true statement. The
solution set is the set of all
real numbers or .
14  9 x   11x   6  2 x  8
17  2 p  p  20    p   5
 3  5
EX 9:
EX 10:
This is a false statement.
 3  5
The solution set is the empty set.

2 y
30   18k  1  9  4  2k 
EX 11:
2
 y  7    2 y 2  4 y  5   12
EX 12:
18
Practice 2-4: More Equations (  and  )
*Do your practice problems on separate paper with plenty of space to show your work. CHECK YOUR
ANSWERS TO THE ODD PROBLEMS USING THE ANSWER KEY PROVIDED!
I. Name the axiom, property, or definition that justifies each statement.

1.
2.
23x  y  
7x  y  z   7( y  x  z)
 23x   2 y 


3.
m  n  p  m  n  p
4.
If x = 3b, and b = y –z, then x = 3(y – z)
5.
If 5 x  3  12, then 5x  3  3  12  3
6.
12  2k  12 
7.
 8  9 
      1
 9  8 
8.
c 3 1  c 3
9.
bcd   bc)(d 
10.
3a  b  0  3a  b
11.
If
12.
 5x y
II.
Simplify and show all work!
1.
3c  c  2  c  1  3  4  c 
2.
2 3a  3  4a  2b    3  2a  3b 
3.
2b  4  b  3  2b  b  b  b  12
4.
1
3  3
5
1 
2
12  a  b  c   6  b  c  a 
2
4  2
6
2 
3
2
3 2
3
b  16, then  b  16 
3
2 3
2
3
2


2k

 17 y 3  1  5 x 3 y 2  17 y 3
19
III.
Solve and check the even answers by substituting your solution into the equation.
1.
x  97  105 17
2.
r  4  (8)  1  (9)
3.
32  y  11
4.
3
 u  24
7
5.
8  y  22  27
6.
2x  5  19
7.
5  y    6  y   5  y   8
8.
4x 1
5
3
9.
5x   2 x  3  6
10.
3 y  1  7 y  3  6 y  14
11.
2 x   x  1  5
12.
8x   3x  4  25
13.
9n  2 1  4n   7
14.
3c   2c  7   17  35
15.
21  10a  4a
16.
3x  2
17.
2  3 10 x   4  5x  2  6
18.
8  a  3  4  2a  1  20
1
6
4
TURN THE PAGE UPSIDE DOWN TO CHECK YOUR ODD ANSWERS WITH THIS KEY:
20
Notes 2-5: More Equations (variables on both sides)
The steps for solving the following equations are the same, but we need get all the variables on one side.
1. 51a  56  44a
4. 2  x  6  3x
2. 71  5x  9x 13
5. 1   2 x  8   x  9  3x
3. 30r  19r  44
6. 8  5  n   2n
21
7.
3
 x  2   12
5
3
8. 21    x  2 
2
9.
9  2y
y
7
11. 3 b  1  b  21  2 1  b 
12.
3  a  2  2a  5a  9
13.
3  a  2  2a  5a  9
14.
3  d  1  8  2  d  3  d  5
10. 6r  2  2  r   4  2r 1
22
Practice 2-5: More Equations (variables on both sides)
*Do your practice problems on separate paper with plenty of space to show your work. CHECK YOUR
ANSWERS TO THE ODD PROBLEMS USING THE ANSWER KEY PROVIDED!
I. Solve each equation and check any five.
1.
98  4b  11b
2.
4n  5  6n  7
3.
1
x5  x
2
4.
1
12  6 x   4  2 x
3
5.
5  2  n   3  n  6
6.
5u  5 1  u   u  8
7.
3  m  5  6  3  m  3
8.
3 5 y  2  y  2  y  3
9.
6r  2  2  r   4  2r 1
10.
3  4  p  2  2 p  3  p  4
11.
3 x  2 1  3  x  2    2 x
12.
6k  1  k   7k  2
13.
5  2m  3  1  2m   2 3  3  2m    3  m  
II.
Name the property or definition illustrated.
1.
a c   1   ac  (a)
2.
(r + 3)9 = 9(r + 3)
3.
If x + y = 2 and y = c, then x + c = 2
4.
If 6 x  3, then x 
5.
1 a  a
6.
6   x  2   3 y    x  2   3 y   6
7.
3   5    7    3 7    5  7 
8.
If 2 y  5  14, then 2 y  19
9.
If (a + b) +0 = x, then (a + b) = x
10.
If
III.
Evaluate each expression with the given variables.
1.
xy  z   2 x  y   if x = -2, y = -3, and z = 5
3.
yz
x 1
2.
1
2
3m
1
1
  , then 3m  
7
14
2
7z 6 y  4x
if x = -2, y = -3, and z = 5
2  x  z 
if x = -2, y = -3, and z = 5
TURN THE PAGE UPSIDE DOWN TO CHECK YOUR ODD ANSWERS WITH THIS KEY:
23
QUIZ REVIEW: Solving Equations
Practice & Self-Assess
1.
x  8  13  2
2.
12  f  17  3
3.
7  12  r   9
4.
2
2 z  32
3
5.
4  k  7   15  3
6.
2b  5  b  1  3  16
7.
3 d  2  2  2  3d   10
8.
42  7  2 x  1
9.
4h 5h

 2
3
3
10.
0  35  5x  2x
24
11.
5  m  2  4  m  1  3  0
12.
2 x  3  2  t   10
13.
2  z  1  3  4 z  2  6 z  0
14.
12 
15.
6 g   3  3g   24
16.
20  8w  40  22
17.
2  z  3  5  z    5  z  7   0
2y
 8
5
TURN THE PAGE UPSIDE DOWN TO CHECK YOUR ANSWERS WITH THIS KEY:
25
Notes 2-6: Solving Equations with Fractions and Decimals
Equations with fractions or decimals look intimidating! But there are some simple tricks to make even the most
daunting equation much easier to solve.
1.
x
 5  2x
3
2. 15  25 
3.
3x  1
 2x  3
7
w
5
The following equations have more two or more fractions with different denominators:
3. 10 
4.
y y
 4
6 3
5x 2 x
11


2
3
6
4x  3  5
5.
2
3
1
1
 6 x  24   20   12 x  72 
4
6. 3
26
The following equations have decimals in them. Think about how decimals can be expressed as fractions.
7. 3b  0.4  0.8
8. 11 f  0.34  0.21
9. 1.7 z  8 1.62z  0.4z  0.32  8
10. x  0.125x 1.2  19.8
11. 0.05 y  0.03  600  y   14
12.
5y  4
y4
 2
9
6
The following equations have both fractions AND decimals.

13.
9
 3  2 z   0.5 6 z  1  2 z   z
4
1
3.2 y  7   4  5 y    0.6  2 y  2 
2
14.
27
Practice 2-6: Equations with Fractions & Decimals
*Do your practice problems on separate paper with plenty of space to show your work. CHECK YOUR
ANSWERS TO THE ODD PROBLEMS USING THE ANSWER KEY PROVIDED!
1. 15 
3.
3
x  12
2
1
 20  4a   6  a
4
2. 8  4d  3d 13
4.
4y  3
9
7
5. 2  3 y  5  9  6  y  3  1
6. 4  r  9  2  12r  14
7. 1.6s  0.4  10
8. 6k 12  2k  2k  4
9. 3 x  2  x  2  x  1
10.
11.
2a  3
 5
4
3
x  14  5
5
12. 
4
 y  4   8
5
1
1
1

13. 12  x    8  x  1
2
3
2

14.
15. 0.02  0.05x  0.3x  0.68
16. 0.8r  0.09  r  3  0.08r
17. 5.36m  0.4m  26.8  0.4m
18. 0.7b  0.15b  5  b   0.2
2
3
 x  6  x  3
3
4
TURN THE PAGE UPSIDE DOWN TO CHECK YOUR ODD ANSWERS WITH THIS KEY:
28
REVIEW: Solving Equations
Work on separate paper and clearly show each step.
I.
Solve each equation. Check your solutions using substitution.
1. 98  4b  11b
5.
2. 4n  5  6n  7
4n  28
 2n
3
6.
3.
1
x5  x
2
1
12  6 x   4  2 x
3
4.
4 y
y
5
7. 5  2  n   3  n  6
8. 5u  5 1  u   u  8
9. 3  m  5  6  3  m  3
10. 3 5 y  2  y  2  y  3
11. 6r  2  2  r   4  2r 1
12. 3  4  p  2  2 p  3  p  4
13. 3 x  2 1  3  x  2    2 x
14. 5  2m  3  1  2m   2 3  3  2m    3  m  
15. 6k  1  k   7k  2
II.
Name the property or definition illustrated.
1.
5  y  2  5 2   y  2
2.
8  4b   8  4 b
3.
1  1  0
4.
b  c  a  a b  c 
5.
            
6.
If 44  6x  8 then 6x  36
7.
 2a  3b 18  18   2a  3b 
8.
 a  b  1  a  b
9.
1
 x  y  1
x y
10.
If 23  2x  11 and x  6 then 23  2  6  11
11.
18 
2
3
 18 
3
2
12.
If
13.
5a   7a 12   5a  7a   12
14.
5 pq  0  13 pq 2  5 pq  13 pq 2
4p
 16 then p  12
3
TURN THE PAGE UPSIDE DOWN TO CHECK YOUR ANSWERS WITH THIS KEY:
29
Notes 2-7: Solving Literal Equations
Do you recognize any of these formulas?
1
A  bh
P  2l  2w
2
A
1
h  b1  b2 
2
C
5
 F  32 
9
S
1. Solve for x: x  a  b
5. Solve for p: 12w  8 p  4u
2. Solve for x: 2 x  3 y  1
6. Solve for w: P  1  2w
3. Solve for y: y  c  2c
4. Solve for h: T  2 rh  2 r 2
7. Solve for g: S 
1 2
gt
2
n
(a  g )
2
8. Solve for h: A    2h
30
9. Solve for k: z  k  3k
10. Solve for x:
11. Solve for h:
a  2 r  r  h 
3 x  2b   9a 15b
31
Practice 2-7: Solving Literal Equations
*Do your practice problems on separate paper with plenty of space to show your work. CHECK YOUR
ANSWERS TO THE ODD PROBLEMS USING THE ANSWER KEY PROVIDED!
I.
Solve for the given variable:
1.
p  abc ; b
2.
m  2s  l ; l
3.
m  2s  l ; s
4.
P  2l  2w ; w
5.
1
A  bh ; b
2
6.
V
7.
A  2 rh ; r
8.
E  mc 2 ; m
9.
A  P  Pr t ; t
10.
2x  3y  6 ; y
11.
bh3
V
;b
3
12.
C
II.
Solve and check evens:
1.
2m  6m 1   3m  2 4m  3
2.
1
1
 49  14 x    12 x  42 
7
6
3.
0.25  0.08  0.4 y   0.2  0.3 y  0.1
4.
5d  2  3  d   4  2d  3  d
5.
1
3.2 y  7   4  5 y    0.6  2 y  2 
2
6.
4 p  5  3 p  11  8 p
Bh
; B
3
Ka  b
;K
a
TURN THE PAGE UPSIDE DOWN TO CHECK YOUR ANSWERS WITH THIS KEY:
32
REVIEW for UNIT 2 TEST
Show each step neatly on separate paper.
CHECK YOUR ANWERS USING THE ANSWER KEY ON THE LAST PAGE!
I.
Simplify:
1. 2  4  3a  2a 2  a 3   3  3a 3  4a  2a 2  1
2.
 4ab  5ab    2a b  ab    5ab  7a b 
2
2
2
2
3. 4  2m   3m  5b    2  b  5  m  2b  
2
1
3
6 x 2  3x  9    5 x 2  10 x  15    8 x 2  16  12 x 
4.

3
5
4
5. 3  2  3 p  2q  4r   3 1  5 p    4  2r   3q  2  
1

1

1

6. 18   5a  2b    12   3a  7b    16   2a  b 
3

2

4

7. (7 x3  y3 )  12( x3  2 y 3 )
8. 5r  3r 2  4 8  3  r  2r 2 
2  1
 1
9. 25   a  b    56a  21b 
5  7
 5
2
1
3 
2
11.  5k  10h    2k  8h   6  k  h 
5
2
2 
3
10.   4  (2 x  y ) 
13. 2  5  a  2b   a   3  a  2b 
14. 12 x  3  2  y  2 x   3( x  y ) 
1
1
 1 
15. 5a    b   3 a  b
2
2
 2 
16. 3  4 y  2  x  3  (2  y ) 
II.
Solve each equation:
1. x  (4)  9  12
4.
7.
10.
13.
16.
6 x  13 x  3 5


2
3
6
2
2
 y4
3
3
 n 3

4
2
6
66 
 a  3
5
2
 p  4  2  p  3  10
5
2
3
 t  6    2  3t 
3
4
1
2

2
3
2. 3  p  6  8
3. h 
5. 0.2a  a  0.8
6. 3  4   5  w
8. 0.3  x  200  0.031000  x   105
11. x  0.4x  8  20
14.
4n  28
 2n
3
12.
9.
6 x
1
2
1
 4  k  2    3  k    4
5
15. 6 x  2  2  x   4  2 x 1
17. 2  y  5  3 3  5 y 
19. x  2   x  8  2 8  3  5  x   x  0
18. 4  5h  9  4 1  h   9  h 11 1
20.
12. 3  7c  2d   2(6d  c)
21. 2   3  y   6
23. 2  n 2  3n  2   n  2n  7   5 24. 
y
2
3
3
3
22. 17  z  12
25.
2
x  7  17
3
33
6 z
5
3
3
29. 21    x  6 
4
27. 1 
26.
3
 v  2   5
4
28. 9  3  2 z  1  18
30. 
III.
Solve for the given variable.
1.
m; F  m  a
2.
5. π; B 
IV.
3.
Name the axiom, property, or definition illustrated.
2
2
 (16)   16
3
3
1
( x  12  y )  z   x  12  y  
z
rs (t )  (t )rs
4.
If x  y  z  a  b , then x  y  z  m  a  b  m
5.
If x  y  z  a  b and y  z  r, then x  r  a  b
6.
7.
d 0  d
d (1)  d
8.
If  2( x  y)  3  4w  3v , then
9.
10.
c  f 
x  ( x)  z  0  z
1.
2.
11.
12.
13.
14.
15.
16.
17.
18.
19.
L
4d 2
2
1
2. a; d  2 at
6. M1; F  G
m1  m 2
d2
6
10 x  15   30
5
3. p; d 
1
p
7. f; v    f
4. m; E  mc 2
8. A;
P
 T 4
A
4w  3v 2( x  y )  3

2
2
m  (n  p)  t   (m  n)  p  t 
2
2
2
2
w  v  z  w  v  z
5
5
5
5
1
1
 45  3    45   3
5
5

a b
 1
b a
(rs)t  r ( st )
If x  9 and 9  y , then x  y
x  (9)  (9)  x
If w 2  7 and 7  5  2 , then w 2  5  2
x  y  ( z )   x  y  z 
34
V.
Multiple Choice
1.
a) 5
Given the two equations 4r  5  17 and 12  2s  2 , what is the product of r and s?
b) 5.5
c) 10.5
d) 25
e) 27.5
2.
The equation: $2,225c-$2,000 = P shows the profit (P) a computer store makes selling (c)
computers. How many computers must the store sell to make a profit of $49,175?
a) 21 computers
b) 22 computers
c) 23 computers
d) 24 computers
e) 25 computers
3.
Which equation has no solution?
a) 5x  12  5  x  2  2
b) 10  3  x  1  3x  14
c) 75x  100  7  x 
d) 3x  4  5  x   3
4.
a) 125
5.
Solve 3.42x 190 1.36x  1.67x 112
b) 140
b) a 2  2b 2
c) 2a 2  b 2  1
d) a 2  2b 2  1
e) 2a 2  2b 2
Circle and identify the errors. Then solve the equation correctly.
14  24   3s    9 s 
14  24  6 s
1.
d) 200
Simplify the expression  3a 2  b 2  1   a 2  b 2  1 .
a) 2a 2
VI.
c) 185
 10  6 s
5
- s
3
2.
9
c  c  3  8
4
9


4  c  c  3   8
4


4c  9c  12  8
 5c  12  8
 5c  4
c
5
4
35
TEST REVIEW ANSWER KEY
I.
1. 7a 3  2a 2  6a  5
5. 63 p  16r  17
9. 3a  7b
2. ab  6ab 2  5a 2b
6. 4a  50b
10. 8 x  4 y
13. 9a  26b
14. 12  15 y  21x
3. 6m  42b
7. 5 x3  31y 3
11. 7k  17h
1
15. 1 a
2
4. 9 x 2  9 x  9
8. 21r 2  7r  32
12. 19c  6d
16. 6 x  15 y  12
II.
1.
7
8. {500}
2.
5
9.
4
15. 
16. 6
22. {29}
29. {90}
23. {-9}
30. {-1}
 1 
3.  
6
10. 6
4. {-2}
5. {1}
12. 3
11. {-20}
 1 
17.  
 13 
24. {-11}
18. 4
19. 1
25. {15}
26. {-9}
7
6. 12
7.
13. 58
14. 14
 66 
20.  
 35 
27. {10}
21. {7}
28. {1}
III.
𝐹
1) 𝐴
2𝑑
2) 𝑡 2
1
3)𝑑
𝐸
4)𝑐 2
𝐿
5) 4𝐵𝑑2
𝐹𝑑2
6)𝐺𝑚
2
𝑉
7)𝜆
𝑃
8) 𝜎𝑡 4
IV.
1) definition of subtraction
2) definition of division
3) commutative property of multiplication
4) addition property of equality
5) substitution
6) identity property for +
7) identity property for x
8) division property of equality
9) closure for addition
10) axiom of opposites (additive inverses)
11) associative for addition
12) distributive property
13) associative for multiplication
14) axiom of multiplicative inverses
15) associative axiom for multiplication
16) transitive property of equality or substitution
17) commutative axiom of addition
18) transitive property of equality or substitution
19) definition of subtraction
V.
1. e
2. c
VI.
5
1.  
3
2. {4}
3. b
4. d
5. e
36