1. No category

1
10. If 23x of x?
(A)
= 8, then what is the value
~
1
(C) 1-­
(B) 1
3
3
(D) 2
11. If 33x = 27 2x value of x?
(A) -1
(B)
then what is the
~
12. If 323 =3 k X
value of k?
(A) 2
1,
3
3k -
(B) 9
(C) 1
3,
(C)
(D) 2
(E) 3
then what is the
10
(D)
13
(E) 20
13. If ab = 3 and ab 2 = 18, then what is
the value of a?
(A)
~
(B) 1
2
(C) 2
(D) 6
(E) 54
14. If x 2 = 25, then x 3 =
(A) -125 only
(B) 125 only
-125 or 125
(E) 15,625
(D) 15
(C)
15. If 5y 2
=
25, then what is the value of
5y 4?
(A) 125
(B) 525
(D) 3,125
(E) 15,625
(C) 625
16. If y2k = 36 when y and k are positive
integers, then what is the value of
y3k?
(A)
V6
(D) 108
(B)
6V6
(C) 54
(E) 216
17.1f3 x = 10 , then 3- 3x =
(A) -1,000
~oo
(D) 1 ,
(B) -27
(E)~
20
1
(C)
1,000
18. If yr x 4 y 2r = 32, then what is the
value ofyT?
(A)~
(B) 2
(D) 4
(E) 8
(C)
VB
19. If 2n = 32, then what is the value of
sn-2?
(B) 25
(E) 3,125
(A) 5
(D) 625
20. If 5x
=
(A) 4
125, then 4 3x (B) 8
5
(C) 16
(C) 125
=
(D) 64
(E) 256
21. If IOn = 1, then 10 2n+ 2 =
(A) 1
(D) 1,000
(B) 10
(E) 10,000
(C) 100
22. If x 7 = a and x 5 = 3m, which of the
following represents m?
(A)
1-2
x
(D) _1_
3x2
(B)~
x2
2
(E)~
3a
23. If c = a 2 b3 and b = a 2d 4 , which of the
following is the correct expression for
c in terms of a and d?
(B) a 7 d 7
(C)
a 7d 12
(E) a 8d 12
24. If yn = x 3n , where n, x, and yare posi­
tive integers, then what is x in terms
ofy?
(A)
0
(D) 3y
(B)
vY
(C)
2::­
3
(E) y3
25. If a 4 = x and b3 = y, then what does
(ab )12 equal in terms of x and y?
(B) x 3y 4
(E) x 12 y 12
pI.
----------------------_J
•
An equation is a statement of equality between two expressions. The solution to
an equation is a value of the variable (a letter used to represent a number) that makes
the equation true. To find the solution, work to get the variable on one side of the
equation and a number on the other side of the equation. Remember, to maintain the
balance of an equation, you must perform the same operation on both sides.
SKILL SET
Match the equation in Column I with its solution in Column II.
Column I
1. 2x-9=-11
A
3
2. -x
+ 100 = 25
B. x
3. 7(x - 3) = 3x - 17
C. x = 100
2 + 1 = -x
1 - 9
4. -x
D. x
4
•
Column II
5
2
x=l
=-1
= -100
Solve for x.
5. 6x + 12 = 4
x
6. -=7
5
8. 3(x + 8) = 2x + 19
9. 2,800 = 14
lOx
7. 5x + 12 = x + 8
2
10. -x
3
4
+ 6 = -x
+4
5
Match the equation in Column I with the equivalent equation in Column II.
Column I
Column II
1
11. -x
+5=1
2
A
12. - 2(5 - x) = 9
B. x + 10 = 2
13. 6x + 15 = 3
C. 2x - 10 = 9
2x+5=1
•
(lZ)
SAT-Type Problems
1. If 3(5)(11)y == 11(3)( -15), then y
(A) -660
(B) -3
(D) 330
(E) 660
(C) 3
2. If -2(5)(6)r = 5(6)(7), then r
(A) -9
(B) -7
-~
(D)
=
(C)
=
-'i
2
(E) 5
7
3. If 2(5)( -7)m
(A) -40
= 5( -7)(8), then m =
(B) -4
(E) 44
(D) 40
(C) 4
2
4. Given that ~ = p, what is the value
of 12p?
4
2
2
(A)~
(B)~
12
3
(D) 4n 2
2
5. If -x = 4p, what is the value of 12p?
6
2
2
(A)~
(B)~
·32
(D) 2x 2
6. If ~
3
(E) 3x 2
= 15, then 3x
=
(A) 5
(B) 12
(D) 45
(E) 135
7. If
254
1,000y
(C)x 2
=
(C) 18
40, then y =
(A) 0.00635 (B) 0.0635
(D) 6.35
(E) 63.5
(C) 0.635
8. If 5,432 = 50 then k =
lOOk
'
(A) 0.10864 (B) 1.0864 (C) 10.864
(D) 108.64 (E) 1,086.4
9. If (x + 5)(8 - 5)
(A) 5
~o.
(B) 10
60, then x
=
(C) 15
If (r + 6)(9 - 3)
=
(D) 20
18, then r
=
(E) 25
=
(A)-9 (B)-3 (C)O (D)3 (E)9
11. If 6(x + 2) - 3x = 8, then x
=
(A)
-~
(B) -±
(C) -~
333
(D)
±3
(E)
12. If 5x - (x - 2)
(A) 2
(B) 3
13. If 4x - 4
(A) 4
(A)
3
18, then x
(C) 4
(D) 5
=
(E) 6
= 14, then x + I =
2
(B) 5
14. If 3x + 1
=
~
=
(D) 18
(C) 7
2
5, then x + -
3
±
(B) 2
3
(E)
(D) 3
(E) 22
=
(C)
2
3
1Q
3
15. If 7 - 5m
= -23, then m -I =
5
(A) 3
(B) 3
(C) 5
5
I
5
(E) 6 I
(D) 5±
5
16. If 3x + 15
1
5
= 33,
(B)
(D) 18
17. If3(2x + 4y)
then x + 5
lOl
3
=
(C)
11
(E) 30
= R,
then 3(4x + 8y)
3R
(A) 2R
(B)
(D) 6R
(E) 9R
(C) 4R
=
'------------­
ax + b = ex - d is an equation with several variables. You can solve for anyone of
the variables by following the procedures covered in the previous section.
SAT-Type Problems
1. If a + b
(A)O
2. If ab
=
a - b, then b
(B)l
(C)a
=
(O)-a
(E)-2a
~, then b =
=
(A) -1 only (B) 1 only (C) -lor 1
(E) -a or a
(0) 0
3. If r = pq, which of the following must
be equivalent to rp?
(A)pq
(0) pr 2
4. If k = am, which of the following must
be equivalent to m 2?
(A)
If;
(0) k
(B)
!
a
(C) kam
2
a
5. If x
= Y,
z
which of the following must
be equal to yz?
(A) xz
(B) xy
(0) xz 2
(E) xyz
(C) xy2
(/J9)
6. If a 3 + b 3
b equal?
=:
a 3 + y3, then what does
(B) y3 + 2a 3
(D)y3 + a 3
) y3
.,» y3
2a3
-
(E) y
7. If 3a = 12C!'., then what does c equal?
4b
c
(C) 4b
(A) 4
(B) 16
(D) 16b
(E) 48ab
4abc
th w h a t IS
.
8 . If - -- 4abcd ,en
5xyz
m
the value of m?
(A) 5dxyz
(B) d
(D) 5xy
(E) 5abc
(C) 5abcd
9. Which expression represents x
if a
=:
(A)
-~
b
(B)_b_
(C)-~
a-c
a
(D) a - c
a+c
+c
(E) a
b
b
10. Ifa(b + 1)
(A)
"* O?
!!.... - c and if x
x
!- -
=:
then b
C,
=
1
(B)
1
1
(E) c - a - I
C -
a
(D)
C -
a
a +a+a
11. I f - 3
and c
(A)
12. If
1
4
c+c+c+c
=:
4'
"* 0, then
(B) 1
a
=:
C
(C)
±
3
(D) 2
a +a +a _ b+b _
3
- -2- -
and c
"* 0, then -ab
C
+
(D)
±
(E) 8
C
+
4
C
+c
=:
C
(A)
~
4
(B) 1
(C) b
6
(E)
l
b
13. The ratio of p: q is 3: 5 and the ratio of
q: r is 4: 9. What is the ratio of p: r?
(A)
1-
(B)
1-
(D)
1-
(E)
~
3
2
(C)
4
~
15
9
14. If the ratio of a: b is 2: 5, and the ratio
of d: b is 3: 2, then what is the ratio of
a:d?
(A)~
(B)
-.£
(E)
~
25
(D)
~
3
(C)~
5
15
2
15. The ratio of a: b is 2: 5, the ratio of
c: d is 5: 2, and the ratio of d: b is 3: 2.
What is the ratio of a: c?
(A)
~
(B)
75
~
(C)
25
1-
(D)
2
~
5
(E) Value cannot be determined from
the information given
. If a + b = c + d
b + c equals
(A) 22
(B) 44
= -a
- d
(C) 66
=
22,
(D) 88
(E) Value cannot be determined from
the information given.
*
17. Ifz 0 and 2x = 3y = 4z, what is the
value of x + y in terms of z?
(A)
~z
(B)!z
(C) 2z
(D)
1Q.z
3 3 3
(E) Value cannot be determined from
the information given.
18. If P
=
2l + 2w, then l=
(A) P - w
(B) P - 2w
2
(D) P
+ 2w
=
2
p + prt, thenp =
(A) A
rt
(B)
(D) rt
(E) rtA
A
+w
(E) 2P - w
2
..... !fA
(C)P
A
1 + rt
(C)~
2rt
'------------------­
To solve a set of equations with two variables, first eliminate one variable. The
method you use to eliminate a variable depends on the equations.
x + y = 15
y = 2x
(1)
(2)
Substitution
Since y = 2x, substitute
2x for y in equation (1).
x
x
+ Y = 15
+ 2x = 15
3x = 15
(1)
x=5
Substitute 5 for x in equation
(2), and solve for y.
y = 2x (2)
y = 2(5)
y = 10
Addition
x + y = 10
x - y =4
(1)
(2)
x + Y = 10
x - y = 4
Adding the two equations
will eliminate one
of the variables.
2x
=
14
x=7
Substitute 7 for x in either
equation, and solve for y
x + Y = 10 (1)
7 + Y = 10
y=3
Subtraction
x
x
+ y = 9 (1)
+ 2y = 11 (2)
x + y = 9
- x - 2y = -11
- y = - 2
Subtracting the two
equations will eliminate
one of the variables.
y=2
Substitute 2 for y in either
equation and solve for x.
x + Y = 9 (1)
x+2=9
x=7
SKILL SET
Solve for x and y.
1.
x
+y
=
y =
28
3x
4.2x + 2y = 16
x = 3y
2.
x
+ y = 32
3.
+ Y = 15
x + 2y = 23
6.
x
3x
x-y=8
5. x + 2y = 11
3x ­ 2y = 1
x
+Y = 9
+ Y = 19
SAT-Type Problems
1. If x
3b and y
=
=
3. If x + Y - z
then x =
2 , what is y
6b + 4
= 8 and x .-
y
+ z = 12,
in terms of x?
(A) _2_
(B) _1_
x+4
x+2
(D) x
2. Ifx 2
+
2
-
7 =y and x
(A) 8
(D) 14,228
(E) 2x
+
(A)2 (B)4 (C)10 (D)20
(E) Cannot be determined from the
information given
(C) 2x
4
4. Ifx + 7 = 2y andy
the value of y?
= 4, then y 2
(B) 80
(E) 14,329
-
1
=
(A)-5
(B)3
=
2x - 1, what is
(C)5
(D)9
(E)13
(C) 81
5. If 2x + y = 6 and x - 6 = y, what is
the value of x?
(A) 0
(B) 2
6. If 5x - 2y
(A)1
(C) 3
= 10
(B)4
(D) 4
(E) 6
= ~, then y =
and x
(C)5 (D)7 (E)9
= ~ and m - a = 80, then what
7 . If ~
5
m
is the value of m?
(A) 2
(B) 20
(D) 120
(E) 200
(C) 80
8. If 3m + 2n = 16 and m - 2n
what is the value of m?
(A) 4
(B) 6
(C) 8
(D) 10
9. If x + y = 4 and x - y
does x equal?
(A) k
+4
(B) 4k
(D) 2
+ 1- k
2
(E) 2
=
=
0, then
(E) 12
k, then what
(C) 4 ­
-1- k
2
k
r + t = sand r - t = -s
10. Based on the equations above, which
of the following must be equal to t?
15. If5x - 4y = 22 and 3x + 6y = 30,
what is the value of x - 5y?
(A) -8
(A) r:s
(B) -r
(D) r
(C) 0
(E) s
(A)6
y -z =x
11. Based on the equations above, which
of the following must be equal to z?
(B) x
(D) -x
(C) y
(E)-y
(C) -2
(D) 0
(E) 2
16. If 2x - y = 0 and x - 2y = 3, then
6x - 6y
x+y=z
(A) 0
(B) -4
=
(B)12
(C)18
(D)21
(E)24
17.1f9x + 2y 2 - 3z 2 = 132 and
9y - 2y 2 + 3z 2 = 12, then x + y =
(A) 3
(B) 8 (C) 16
(D) 120
(E) 144
18. If a, b, and c are positive integers and
if a = 2b, and a 2 + b 2 = c, which of the
. following cannot equal c?
10
11
x
y
+12
32
(A) 5
(B) 20
(D) 125
(E) 500
(C) 50
9
8
x
z
+7
37
12. In the correctly worked addition prob­
lems above, what is the value of
z -y?
(A) -14
(B) -12
(D) 12
(E) 14
13. If x + y = 12 and x - y
the value of x 2 + y2?
(A) 20
(B) 24
(D) 104
(E) 208
(C) 5
8, what is
=
(C) 100
14. If8x - 4y = 20 and 4x - 8y
is the value of x - y?
(A) 1
(B) 2
(C) 3
(D) 4
=
4, what
(E) 6
I~
J
l.­
The absolute value of a number is its distance from zero. Symbolically, the absolute
.Jue of x is written as Ix I. Thus, 131 = 1-31 = 3. Both 3 and -3 are 3 units from zero
on the number line.
When you solve absolute value equations, you will consider two possibilities:
1. The expression inside the absolute value sign is positive.
2. The expression inside the absolute value sign is negative.
Each equation has possible solutions for both cases:
Example Solve Ix +
Case 1:
51 = 12
x + 5 is positive. The expression is therefore equal to 12.
Simply drop the absolute value sign and solve.
x + 5 = 12
x=7
Case 2:
x + 5 is negative. The negative of the expression is now equal to 12.
Exchange the absolute value sign for a parenthesis preceded by a nega­
tive sign.
-(x
+
5)
= 12
x + 5 = -12
x = -17
Answer: x = 7 or x = -17
SKILL SET
l.What is the sum of the solutions to the equation 1-5x - 51 = 45
(A) -18
(B) -9
(C) -2
(D) 2
(E) 18
2. -12 and 24 are solutions to the which of the following equations?
(A)
(C)
(E)
I-x - 61 = 18
Ix + 61 = 6
Ix - 61 = 18
(B)
(D)
I-x + 61 = 30
Ix + 61 = 30
3. What is the product of the solutions to the equation
(A) -30
(B) 24
C) 30
(D) 48
~ + 6 = 4?
(E) 180
(~
4. What is the smaller solution to the equation 18 - ~
(A) 10
(B) 14
(C) 22
(D) 56
=
4
(E) 88
5. At Family Math Day, first-grade students won a prize for guessing within 8
the number of marbles in the jar. If the jar contained 44 marbles, how many
winning numbers are a multiple of 3?
(A) 2
(B) 3
(C) 4
(D) 6
(E) 14
6. What is the larger solution to the equation /6 (A) 17
(B) 30
(C) 60
(D) 72
~ I~ 11?
(E) 102
7. What is the difference between the solutions to the equation
(B) 10
(A) 7
(C) 12
(D) 18
SAT-Type Problems
1. What is the product of the solutions of
the equation 13x + 211 = 9?
(A) -49
(B) -40
(D) 40
(E) 49
(C) -9
2. -6 and 14 are solutions to which of
the following equations?
(C)
Ix + 41
Ix - 21
(E)
Ix
(A)
10
(B)
= 12
(D)
=
- 161
=
Ix - 41 = 10
Ix + 161 = 10
10
3. Which is the greater solution for the
equation I -3x - 121 = 30?
(A) 6
4. If x
(B) 10
(C) 14
= -2, what is
(D) 15
the value of
IxO I - Ixl I + Ix 2
1
(A) -6
(B) -5
(D) 14
(E) 15
­
Ixsl ?
(C) -3
(E) 18
(E) 30
12 + ~ I=
5?
5. x = 3 is the complete solution set to
which of the following equations?
(A) x - 6 = 9
,J)
(E)
(B) -2x = -6
I-x + 31
Ix I = 3
=
6
(D)
14x - 61
=
6
6. If Jorge scores within 3 points of Sam,
they will be placed on opposing teams.
Sam scores 12 points and they play on
opposing teams. How many different
point totals could Jorge score?
(A) 9
(B) 8
(D) 6
(E) 3
(C) 7
7. Prizes are awarded to all participants
in a charity event who guess within 5
the number of people attending the
event. If 43 people attended the event,
how many guesses are a multiple of
3?
(A) 5
(B) 4
(D) 2
(E) 1
(C) 3
:~t What is the sum of the solutions to
...•. the equation I -3x + 181 = 6?
(A) 4
(B) 8
(D) 16
(E) 20
(C) 12
: 9.4 and 8 are solutions to which of the
following equations?
Ix + 21
(C) Ix + 61
(E) Ix - 61
(A)
6
(B)
= 10
(D)
=
Ix - 2/ = 2
Ix + 161 = 10
= 2
10."-- ich is the smaller solution for the
equation 116 - 4x I = 8?
(A) -6
(D) 2
. (B)-4
(E) 4
(C) -2
Part 2: Equations
<
< ••
,
<;1. What is t~e sum of the solutions to
.... the equatIOn 1-3x + 181 = 6?
(A) 4
(B) 8
(D) 16
(E) 20
175
10. Which is the smaller solution for the
equation 116 - 4x I = 8?
(C) 12
(A) -6
(B) -4
(D) 2
(E) 4
(C) -2
9. 4 and 8 are solutions to which of the
following equations?
."(
(A)
Ix + 21
(C)
Ix+61 =10
(E)
Ix - 61
=6
(B)
Ix - 21 = 2
(D)
Ix + 161
=
10
= 2
'----------------------­
Binomials are algebraic expressions with two terms.
+y
3x
x 2 - y2
4a 2b - 1
You can use the distributive property or FOIL to multiply binomials.
• Using the distributive property, multiply each term of one binomial by each term of
the other.
(x
+
5)(x
+
3)
= x(x + 3) + 5(x + 3)
= x2
+
3x
+
5x
+ 15
x 2 + 8x + 15
=
• Using FOIL,
First L
st
rr=ri
+
+
(x
5)(x
u
t
3) = x 2
t
+
F
~
3x
+
+ 15
I
L
5x
a
Outer
=
x 2 + 8x + 15
The product x 2 + 8x + 15 is a quadratic expression. A quadratic expression is an
expression of the second degree.
You can factor a number or a quadratic expression by breaking it up into its factors.
<
12
'---y---'
=
3
X
4
'----v--'
product factors
x2
+ 2x
= x(x
+ 2)
'----r--'
'--r-'
product
factors
x 2 + 8x + 15 = (x + 3)(x + 5)
'--y---1
'--y---1
product
factors
176
f
Category II: Algebra and Functions
Strategy Note
Quadratics that occur often on the SAT are listed below. Memorize the
different forms each quadratic can take.
1. (x
+ y)2 =
(x
+ y)(x + y) = x 2 + 2xy + y2
f
+ 3)2 = x 2 + 6x + 9
1. (x
2. (x - y)2 = (x - y)(x - y) = x 2 - 2xy + y2
2. (x - 3)2 = x 2 - 6x + 9
3. (x + y)(x - y) = x 2 - y2
3. (x + 3)(x - 3) = x 2 - 9
SKILL SET
Find the product or the square.
1. 3(x + 2)
2. y (y - 1)
3. (z + l)(z - 3)
5. (x - 1)2
6. (y + 2)2
7.
9. (p + q)(P - q) 10. (a + b)2
(a - 3)(a
+
4.
(b - 2)(b - 1)
8. (c - d)2
3)
11. (2k - 3)2
12. (2x + y)2
15. x 2 -x
16. m 2 + 2m + 1
19. x 2 + 6x + 9
20. a 2
Factor.
13. 2p + 4q
14. b 2
17. s2 - t 2
18. y2 - 2y
+
3b
+
1
MODEL QUESTIONS
Model 1 If x 2 + y2
(A) -13
= 37 and xy = 24, what is the value of (x
(B) -11
(C) 11
(D) 13
- y)2?
(E) 61
Solution: Recognize that x 2 + y2 and xy can be found in (x - y)2.
Rewrite.
(x - y}2 = (x - y)(x - y)
Multiply using the FOIL method.
= x2 -
Rearrange the terms to get x 2 + y2.
= x 2 + y2
Sub-in the given values, and evaluate.
= 37 - 2(24) = -11
Answer: B
2xy
+ y2
- 2xy
-
8a
+ 16
I
r
If':·,
Part 2: Equations
= 14N3, x + Y = 21N, and N *- 0, what is x
Model2 If x 2 - y2
(A) .Jv2
~2
(B)
(C)
3~2
- y in terms of N?
(E) 14N 3 + 21N
(D) 7N2
x 2 - y2 = 14N3
Solution:
Factor the quadratic expression.
Sub-in 21N for (x
(x
+ y)(x - y) = 14N 3
21N (x - y) = 14N 3
+ y).
3
2
14N
(x-y)=
- - =2N
-­
21N
3
Divide both sides by 21N.
Answer:
177
B
SAT-Type Problems
1. If x 2
+ 81
(A) -81
=
6 x 27, then x could be
(B) -9
(C) 0
(D) 3
(E) 14
2. If x = v'5 and y = 0, what is the
value of (x + y) (x - y)?
(A) -2
(D)
(B)
V35
V2
(C) 2
(E) 35
3. (3 + ab)(7 - ab) =
(A) 21 - a 2 b2
(C) 21
+ 4ab
+ 4ab - a 2 b 2
(E) 21 - ab
4. (5x - 3y)2 + (5x + 3y)2 =
..l.lI·
..
(B) 16
(C) 8
(C) 13
(D) 16
(E) 26
(B) 0.1
(C) 1 (D) 7 (E) 10
8. If x 2 + 7x + 8 = (x + 3)(x + 4) + p,
thenp =
(B) 4
(C) 2
(D) -2
(E)-4
9. If x 2 - 3x - 2 = 0, then what is the
value of 2x 2 - 6x - 11?
(B) 50x 2 + 18y2
(C) 50x 2 - 30xy + 18y2
(D) 50x 2 + 15xy + 18y2
(E) 50x 2 + 30xy + 18y2
(A) 36
(B) 7
7. If 4x+3y
2
2 -- 10, w hat
at iIS t h e
16x - 9y
value of 4x - 3y?
(A) 8
(A) 100x2
5. If x + y = 12 and x - y
x2 _ y2 =
(A) 3
(A) 0.01
(B) 21
(D) 21
x 2 + 5x + 6
(
)
x+2
rounded to the nearest whole
number is
6. If x = 10.00001, then
(A) -9
(B) -7
(C) 0
(D) 7
(E) 9
10. (2x + 3y)2 - (2x - 3y)2 =
1 then
= -,
3
(A) 12x
(D)
(D) 4
12xY
(B) 12y
(C) 24x
(E) 24xy
(E) 2
.....
L.-------------_J
Rational Expressions are handled the same way that fractions are handled in
arithmetic.
.
Note, however, that just as fractions are meaningless when the denominator equals
zero, the rational expression is meaningless when the denominator equals zero. We can
never divide by zero.
.1 is meaningless when a =
a
7
mix - 2)
0.
is meaningless when m
when x = 2.
SKIl-L SET
Simplify.
a + b
1.
a-b
a-b
4. ~+~
n
m
(x + y)2 .
7.
z3
Z
x+y
b
b+a
2.
a
a+b
5.
4x
+ 3x
5
7
8.
3p2 -'- 6p 3
----;:'2' 2r 3
3.
m
r-m
6.
c3 d 2
. c2
d5
9.
a-b
2
r
r - m
(a - b)2
6
Solve for x.
1
2
10. 1+-=3
x
11.
1
x+2
1
--
1 2
12. -+-=1
x
x
3
Strategy Note
Remember that the answer choices in the SAT may not be written in the
form in which you have done your calculations. For example, your answer may be
m
+ x , while the answer choice may be
x
m
x
+ 1.
= 0, or
SAT-Type Problems
1.
4a-~
3
2
-.!
(A) 2a
3
(C) 2a - ~
3
(B) 2a - 1
1 (E) 8a - .!
·33
(D) 2a -
x3 . x4 . x5
6
= x n , then n =
x
(A) 2 (B) 6 (C) 10 (D) 54
2. If
(E) 66
3. If ~ = 1, then:L + 5 =
y
5
x
(A)
(B)~
1
6
(C)
26
(D) 26
~
5
(E) 10
5
(~y
4. If a=/;O, then (:
(A) --.L
(B)
16
5 If x
•
a3
= -
b
8
=
~1_
b - x
(B)
1
(C)
and a . b
(A) bx
(D)
1
Y
4
=/;
~
(D)
~
16
(E)
s.
4
1
0 then - 3 =
'a
(C) ~
x
b
(E)--.L
bx
6. If g and m are positive integers, then
which of the following must be equal
to .fL..?
m
+ g (C) g + m
m+m
m+g
(A)L+ 5
m +5
(B) g
(D) g - m
(E)L
gm
2
m2
7. If x + 2x + 3x + 4x = 9, then x =
x·x
9
1.)10
(B) 1
(E) 10
(D) 5
8. If...!!....- = ~, then which of the
25
n
following could equal 50?
(A) n 2
(D)
(B) 50n
25 2
n2 -
(C) 2n - 25
(E) 3n - 25
2
9. If -2
+2+
- +2
2 + - + 2 = 0, then x
x
x
x
(A) -2
(B) -1
(D) 0
(E) 1
(C)
-.l
2
10. I f _8_ =
8 ,then what is the
a+l
2a-2
.
value of a?
.) 3
(C) 1
(B) 2
(E)
(D) 0
-.l
3
2 - = --3
11. I f - , then what is the
y+2
y-3
value ofy?
(A) -3
(B) -2
(D) 2
(E) 3
12. If a 2 =
l, then c2 =
C
(A)
l
a
(C) 0
=
13. If m
2
9
= Sr
and r = lOt,
then what
(A)~m
(B)
(D)~m
(E)~m
!:-m
5
(C)~m
10
14. If rand s are positive integers and
1- + s =
1, then what is the value of s
r
(A) x = a (D) x> 1
9
2
b
a
then which of the following state­
ments must be true?
is t in terms of m?
25
.
1
1
17. If 0 < a < 1 < b and x = - - -,
b
(B) x> 0
(E) 0
(C) x = 1
<x <2
18. If 1 + x = 1, then which of the
g
following is an expression for x in
terms of g?
in terms of r?
(A)_I_
(B) 1 - r
(A) r - 1
r
(C) 1 - r
1+r
r
(D)_r_
(E)_r_
1+r
r - 1
r
m
i
*" 0,
then what is the value of y?
(c)2
3m 3
(D) m
2 y , then what is
6
the ratio of
x toy?
(A) 6 to 5
(D)
(E)g - 1
(A) -2 and -6 (B) -3 and -4
(C) 2 and 6
(D) 3 and 4
(E) 1 and 12
1
1
1
20. Solve - + - = ­
p2 18
2p
(E) 3m 2
3
6
(C) g - 1
g
m
2
16. If ~x =
(B)I-g
19. Solve 1- + ~2 = ~
4
m
-rn
rri
(A)_1_
(D) - g -
g-1
h
15. If m+m+m - y = 1, were
m
m
g -:-1
5 to 7
to 7
(E) 7 to 5
(B) 6
(C) 7 to 6
(A) -3 and -6 (B) -3 and 6
(C) -2 and 9
(D) 3 and -5
(E) 3 and 6
g
-------------------------A radical equation is an equation with at least one radical expression containing a
variable under the radical symbol. To solve a radical equation, use the following power
I'ule:
If x = y and n is a positive integer, then x n
= y",
Also recall that (~r = x.
SAT-Type Problems
All ofthe following problems are designed as Student-Produced Response Questions.
1. If vx=3 = 3, then what is the value
of x?
4. If v5x - 1 + 2 = 10, then what is the
value of x?
2. If v3x - 8 = 4, then what is the value
of x?
5. If v6b + 4 = 20, then what is the
value of x?
3. If 3v2x + 1 = 15, then what is the
value of x?
6. Ifv3x + 3 = 3vx-1, then what is the
value of x?
7. If Y20x = 2Y4x + 5, then what is the
value of x?
9. If 2Y6X = V4X2, then what is the
value of x?
8. If 5Y5x - 1 = 7V2x + 5, then what is
the value of x?
10. If Y3x + 1 = x - 3, then what is the
value of x?
~
------------------
-----J
An inequality states that one quantity is less than (or greater than) another.
-2 < 3
x+2>x+l
T
T
is less than
is greater than
SKILL SET
Match the expression in column I with the phrase in column II.
Column II
Column I
1.
A. k is less than O.
O<k<1
B. k is greater than
2. k:> - I '
-1 but less than
or equal to O.
c. k is not equal to O.
3. k<O
4. -1 < k
<
D. k is greater than 0 but less than 1.
0
E. k is greater than
5. k=l=O
or equal to -l.
Solve the inequality.
6. 3x - 8
< 1
7. - 5x + 1 > - 9
8. 1 < 2a + 3 < 7
--------------------------------SAT-Type Problems
1. If 10 - x> 7, then x Can be any of
these numbers except
. (A) -5
(D) 0
(B) -3
(C) -2
(E) 3
2. If .!Q. < x, then which of the following
x
values could be x?
(A) -10 (B) -5 (C) 1 (D) 2 (E) 5
3. If 0 < x < 1., which of the following
2
statements must be true?
(A) 2x = 1
(D) x
< x2
(B) x> x 2
(E) 2x
> 1
x2
2
(C)x = ­
-
4. If -1 < x < 0, then which of the
following statements must be true?
(A) 2x
(D) x
= -1
< x2
x2
2
(B) x> x 2
>
(E) 2x
(C) x = ­
1
(B) 12 < rt < 26
< rt < 15
(D) 35 < rt < 165
(C) 35 < rt < 55
(E) 55 < rt < 105
6. If a > b, c < b, and d > a, then which
of the following is the correct
relationship?
(E) c
< b<a <d
<a < c<d
<a < d < b
(B) c
(D) b
< b< d <a
<c <a <d
(A) x
< 1-
(D) x
< -1- (E) x > 1
8. If K
> 1-
(C) x
>
-!
3 3 3
3
= - ~, then which of the
following inequalities is correct?
K < K3 < K2
(D) K3 < K2 < K
(B)
< m2 < m3
(C) m 2 < m 3 < m
.
2
(E) m 3 < m < m
(A) m
=0
(C) x
<
-II
2
11. If ~ > x, then which of the following
6
values could be x?
(A) 6
(B) 3
(D) -3
(E) Cannot be determined
(C) 0
12. If a and b are integers, and a < b < 0,
which ofthe following must be true?
I. aXb>O
II. -b>-a
7. If 1 - 3x < 0, then which of the
following contains all the values for x?
(B) x
(B) x
2
(A) 7
(C) b
(A) x> 0
(D) x > 11- (E) x <
5. If 7 < r < 11 and 5 < t < 15, then
(A) c
10. If3 - x < 3x + 3, which of the
following must be true?
III. 0 <
l
a
< 1
II only
(C) I and III only (D) II and III only
(E) I, II, and III
(A) I only
(B) I and
17. If x > 1, then which of the following
decreases as x increases?
.L3. Ifx > y, and xy < 0, which of the
following inequalities must be true?
I. x> 0
I.
II. y > 0
II.
III. -1­
III. ~< 0
x+1
y
(A) I only
(B) II only
(C) III only (D) I and III only
(E) I, II, and III
(A) I only
(B) II only
III only (D) I and II only
(E) I and III only
(C)
14. If 1 :5 x :5 3 and 3 :5 Y :5 5, the
18. What are all the values of x for which
(x - 3)(x + 5) < O?
least possible average of Ix and Iy is
-_.
(A)
(B)~
--.£
(C)
15
15
(E)
(D) .!
3
> - 5 (B) x > 5
(C) x < 3
(D) -3 < x < 5
(E) -5 < x < 3
(A) x
1.
3
~
3
19. What are all the values of x for which
(x + 6)(x + 3) < O?
15. If 1 :5 X :5 3 and 3 :5 Y :55, the
greatest possible average of I and 1.. is
x
y
(A)
--.£
15
(B)
1
x2
3x 2 - x
~
15
(C)
1.
3
(D).!
3
(E)
~
3
16. If a and b are integers and
a + b > a - b, which of the following
must be true?
(A) a
< 0
(B) b
< 0
(D) b
> 0
(E) b
> a
(C) a = b
< x < 3 (B) -6 < x < -3
x < 3
(D) x < -3
(A) -6
(C)
(E)x <-6
20. If x 2 - 3x + 4 > x 2 + 3x + 4, then
which of the following best describes
x?
< 0
(B) x = 0
(D) x < -3 (E) x > 3
(A) x
(C) x = 1