GMAT exam success Episode 2 Part 8 pps

3. If you are subtracting, change the subtraction sign to addition, and change the sign of the number fol-
lowing to its opposite. Then follow the rules for addition:
a. –5 + –6 = –11 b. –12 + (+7) = –5
Remember: When you subtract, you add the opposite.
M
ULTIPLYING AND DIVIDING INTEGERS
1. If an even number of negatives is used, multiply or divide as usual, and the answer is positive.
a. –3 × –4 = 12 b. (–12 Ϭ –6) × 3 = 6
2. If an odd number of negatives is used, multiply or divide as usual, and the answer is negative.
a. –15 Ϭ 5 = –3 b. (–2 × –4) × –5 = –40
This is helpful to remember when working with powers of a negative number. If the power is even,
the answer is positive. If the power is odd, the answer is negative.
Fractions
A fraction is a ratio of two numbers, where the top number is the numerator and the bottom number is the
denominator.
R
EDUCING FRACTIONS
To reduce fractions to their lowest terms, or simplest form, find the GCF of both numerator and denominator.
Divide each part of the fraction by this common factor and the result is a reduced fraction. When a fraction
is in reduced form, the two remaining numbers in the fraction are relatively prime.
a. b.
When performing operations with fractions, the important thing to remember is when you need a com-
mon denominator and when one is not necessary.
ADDING AND SUBTRACTING FRACTIONS
It is very important to remember to find the least common denominator (LCD) when adding or subtract-
ing fractions. After this is done, you will be only adding or subtracting the numerators and keeping the com-
mon denominator as the bottom number in your answer.
a. b.
6
15
ϩ

10
15
ϭ
16
15
3
ϫ x
y ϫ x
ϩ
4
xy
ϭ
3x ϩ 4
xy
2
ϫ 3
5 ϫ 3
ϩ
2 ϫ 5
3 ϫ 5
LCD ϭ xyLCD ϭ 15
3
y
ϩ
4
xy
2
5
ϩ
2

3
32x
4xy
ϭ
8
y
6
9
ϭ
2
3
– ARITHMETIC–
329
MULTIPLYING
FRACTIONS
It is not necessary to get a common denominator when multiplying fractions. To perform this operation, you
can simply multiply across the numerators and then the denominators. If possible, you can also cross-can-
cel common factors if they are present, as in example b.
a. b.
DIVIDING F
RACTIONS
A common denominator is also not needed when dividing fractions, and the procedure is similar to multi-
plying. Since dividing by a fraction is the same as multiplying by its reciprocal, leave the first fraction alone,
change the division to multiplication, and change the number being divided by to its reciprocal.
a. b.
Decimals
The following chart reviews the place value names used with decimals. Here are the decimal place names for
the number 6384.2957.
It is also helpful to know of the fractional equivalents to some commonly used decimals and percents,
especially because you will not be able to use a calculator.

0.4 ϭ 40% ϭ
2
5
0.3 ϭ 33
1
3
% ϭ
1
3
0.1 ϭ 10% ϭ
1
10
T
H
O
U
S
A
N
D
S
H
U
N
D
R
E
D
S
T

E
N
S
O
N
E
S
D
E
C
I
M
A
L

P
O
I
N
T
T
E
N
T
H
S
H
U
N
D

R
E
D
T
H
S
T
H
O
U
S
A
N
D
T
H
S
T
E
N
T
H
O
U
S
A
N
D
T
H

S
638
42
95
7
.
3x
y
Ϭ
12x
5xy
ϭ
3
1
x
1
y
1
ϫ
5xy
1
12
4
x
1
ϭ
5x
4
4
5

Ϭ
4
3
ϭ
4
1
5
ϫ
3
4
1
ϭ
3
5
12
25
ϫ
5
3
ϭ
12
4
25
5
ϫ
5
1
3
ϭ
4

5
1
3
ϫ
2
3
ϭ
2
9
– ARITHMETIC–
330
ADDING AND SUBTRACTING DECIMALS
The important thing to remember about adding and subtracting decimals is that the decimal places must be
lined up.
a. 3.6 b. 5.984
+5.61 –2.34
9.21 3.644
MULTIPLYING DECIMALS
Multiply as usual, and count the total number of decimal places in the original numbers. That total will be
the amount of decimal places to count over from the right in the final answer.
34.5
× 5.4
1,380
+ 17,250
18,630
Since the original numbers have two decimal places, the final answer is 186.30 or 186.3 by counting over
two places from the right in the answer.
DIVIDING DECIMALS
Start by moving any decimal in the number being divided by to change the number into a whole number.
Then move the decimal in the number being divided into the same number of places. Divide as usual and

keep track of the decimal place.
.3
5.1ͤෆෆෆෆෆ1.53
⇒
51ͤෆෆෆෆෆ15.3
Ϫ15.3
0
1.53 Ϭ 5.1
0.75 ϭ 75% ϭ
3
4
0.6 ϭ 66
2
3
% ϭ
2
3
0.5 ϭ 50% ϭ
1
2
– ARITHMETIC–
331
Ratios
A ratio is a comparison of two or more numbers with the same unit label. A ratio can be written in three ways:
a: b
a to b
or
A rate is similar to a ratio except that the unit labels are different. For example, the expression 50 miles
per hour is a rate
—

50 miles/1 hour.
Proportion
Two ratios set equal to each other is called a proportion. To solve a proportion, cross-multiply.
Cross multiply to get:
Percent
A ratio that compares a number to 100 is called a percent.
To change a decimal to a percent, move the decimal two places to the right.
.25 = 25%
.105 = 10.5%
.3 = 30%
To change a percent to a decimal, move the decimal two places to the left.
36% = .36
125% = 1.25
8% = .08
Some word problems that use percents are commission and rate-of-change problems, which include
sales and interest problems. The general proportion that can be set up to solve this type of word problem is
, although more specific proportions will also be shown.
Part
Whole
ϭ
%
100
x ϭ 12
1
2
4x
4
ϭ
50
4

4x ϭ 50
4
5
ϭ
10
x
a
b
– ARITHMETIC–
332
COMMISSION
John earns 4.5% commission on all of his sales. What is his commission if his sales total $235.12?
To find the part of the sales John earns, set up a proportion:
Cross multiply.
RATE OF CHANGE
If a pair of shoes is marked down from $24 to $18, what is the percent of decrease?
To solve the percent, set up the following proportion:
Cross multiply.
Note that the number 6 in the proportion setup represents the discount, not the sale price.
SIMPLE INTEREST
Pat deposited $650 into her bank account. If the interest rate is 3% annually, how much money will she have
in the bank after 10 years?
x ϭ 25% decrease in price
24x
24
ϭ
600
24
24x ϭ 600
6

24
ϭ
x
100
24 Ϫ 18
24
ϭ
x
100
part
whole
ϭ
change
original cost
ϭ
%
100
x ϭ 10.5804 Ϸ $10.58
100x
100
ϭ
1058.04
100
100x ϭ 1058.04
x
235.12
ϭ
4.5
100
part

whole
ϭ
change
original cost
ϭ
%
100
– ARITHMETIC–
333
Interest = Principal (amount invested) × Interest rate (as a decimal) × Time (years) or I = PRT.
Substitute the values from the problem into the formula I = (650)(.03)(10).
Multiply I = 195
Since she will make $195 in interest over 10 years, she will have a total of $195 + $650 = $845 in her
account.
Exponents
The exponent of a number tells how many times to use that number as a factor. For example, in the expres-
sion 4
3
, 4 is the base number and 3 is the exponent,or power. Four should be used as a factor three times: 4
3
= 4 × 4 × 4 = 64.
Any number raised to a negative exponent is the reciprocal of that number raised to the positive expo-
nent:
Any number to a fractional exponent is the root of the number:
Any nonzero number with zero as the exponent is equal to one: 140° = 1.
Square Roots and Perfect Squares
Any number that is the product of two of the same factors is a perfect square.
1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9, 4 × 4 = 16, 5 × 5 = 25,
Knowing the first 20 perfect squares by heart may be helpful. You probably already know at least the
first ten.

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400
256
1
4
ϭ
4
2 256 ϭ 4
27
1
3
ϭ
3
2 27 ϭ 3
25
1
2
ϭ 2 25 ϭ 5
3
Ϫ2
ϭ 1
1
3
2
2
ϭ
1
9
– ARITHMETIC–
334
Radicals

A square root symbol is also known as a radical sign. The number inside the radical is the radicand.
To simplify a radical, find the largest perfect square factor of the radicand
͙ෆ32 = ͙ෆ16 × ͙
ෆ
2
Take the square root of that number and leave any remaining numbers under the radical.
͙ෆ32 =
4͙
ෆ
2
To add or subtract square roots, you must have like terms. In other words, the radicand must be the
same. If you have like terms, simply add or subtract the coefficients and keep the radicand the same.
Examples
1. 3͙
ෆ
2 + 2͙
ෆ
2 = 5͙
ෆ
2
2. 4͙
ෆ
2
–
͙
ෆ
2 =
3͙
ෆ
2

3. 6͙
ෆ
2 + 3͙
ෆ
5 cannot be combined because they are not like terms.
Here are some rules to remember when multiplying and dividing radicals:
Multiplying: ͙
ෆ
x × ͙
ෆ
y = ͙
ෆෆෆ
xy
͙
ෆ
2 × ͙
ෆ
3 = ͙
ෆ
6
Dividing:
Counting Problems and Probability
The probability of an event is the number of ways the event can occur, divided by the total possible outcomes.
The probability that an event will NOT occur is equal to 1 – P(E).
P1E2ϭ
Number of ways the event can occur
Total possible outcomes
B
25
16

ϭ
2 25
2 16
ϭ
5
4
B
x
y
ϭ
2 x
2 y
– ARITHMETIC–
335
The counting principle says that the product of the number of choices equals the total number of pos-
sibilities. For example, if you have two choices for an appetizer, four choices for a main course, and five choices
for dessert, you can choose from a total of 2 × 4 × 5 = 40 possible meals.
The symbol n! represents n factorial and is often used in probability and counting problems.
n! = (n) × (n – 1) × (n – 2) × × 1. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.
Permutations and Combinations
Permutations are the total number of arrangements or orders of objects when the order matters. The formula
is , where n is the total number of things to choose from and r is the number of things to
arrange at a time. Some examples where permutations are used would be calculating the total number of dif-
ferent arrangements of letters and numbers on a license plate or the total number of ways three different peo-
ple can finish first, second, and third in a race.
Combinations are the total number of arrangements or orders of objects when the order does not mat-
ter. The formula is , where n is the total number of objects to choose from and r is the size
of the group to choose. An example where a combination is used would be selecting people for a commit-
tee.
Statistics

Mean is the average of a set of numbers. To calculate the mean, add all the numbers in the set and divide by
the number of numbers in the set. Find the mean of 2, 3, 5, 10, and 15.
The mean is 7.
Median is the middle number in a set. To find the median, first arrange the numbers in order and then
find the middle number. If two numbers share the middle, find the average of those two numbers.
Find the median of 12, 10, 2, 3, 15, and 12.
First put the numbers in order: 2, 3, 10, 12, 12, and 15.
Since an even number of numbers is given, two numbers share the middle (10 and 12). Find the aver-
age of 10 and 12 to find the median.
The median is 11.
10 ϩ 12
2
ϭ
22
2
2 ϩ 3 ϩ 5 ϩ 10 ϩ 15
5
ϭ
35
5
n
C
r
ϭ
n!
r!1n Ϫ r 2!
n
P
r
ϭ

n!
1n Ϫ r 2!
2
– ARITHMETIC–
336
Mode is the number that appears the most in a set of numbers and is usually the easiest to find.
Find the mode of 33, 32, 34, 99, 66, 34, 12, 33, and 34.
Since 34 appears the most (three times), it is the mode of the set.
NOTE: It is possible for there to be no mode or several modes in a set.
Range is the difference between the largest and the smallest numbers in the set.
Find the range of the set 14, –12, 13, 10, 22, 23, –3, 10.
Since –12 is the smallest number in the set and 23 is the largest, find the difference by subtracting them.
23 – (–12) = 23 + (+12) = 35. The range is 35.
– ARITHMETIC–
337

Translating Expressions and Equations
Translating sentences and word problems into mathematical expressions and equations is similar to trans-
lating two different languages. The key words are the vocabulary that tells what operations should be done
and in what order. Use the following chart to help you with some of the key words used on the GMAT® quan-
titative section.
SUM
MORE THAN
ADDED TO
PLUS
INCREASED BY
PRODUCT
TIMES
MULTIPLIED BY

QUOTIENT
DIVIDED BY
EQUAL TO
TOTAL
DIFFERENCE
LESS THAN
SUBTRACTED FROM
MINUS
DECREASED BY
FEWER THAN
Ϭ
ϩ
Ϫϫ
ϭ
CHAPTER
Algebra
21
339
The following is an example of a problem where knowing the key words is necessary:
Fifteen less than five times a number is equal to the product of ten and the number. What is the
number?
Translate the sentence piece by piece:
Fift
ee
n less than five times the numb
er equals the pro
duct of 10 and
x.
5x – 15 = 10x
The equation is 5x – 15 = 10x

Subtract 5x from both sides: 5x – 5x – 15 = 10x – 5x
Divide both sides by 5:
–3 = x
It is important to realize that the key words less than tell you to subtract from the number and the key
word product reminds you to multiply.

Combining Like Terms and Polynomials
In algebra, you use a letter to represent an unknown quantity. This letter is called the variable. The number
preceding the variable is called the coefficient. If a number is not written in front of the variable, the coeffi-
cient is understood to be one. If any coefficient or variable is raised to a power, this number is the exponent.
3x Three is the coefficient and x is the variable.
xy One is the coefficient, and both x and y are the variables.
–2x
3
y Negative two is the coefficient, x and y are the variables, and three is the exponent of x.
Another important concept to recognize is like terms. In algebra, like terms are expressions that have
exactly the same variable(s) to the same power and can be combined easily by adding or subtracting the coef-
ficients.
Examples
3x + 5x These terms are like terms, and the sum is 8x.
4x
2
y + –10x
2
y These terms are also like terms, and the sum is –6x
2
y.
2xy
2
+ 9x

2
y These terms are not like terms because the variables, taken with their powers,
are not exactly the same. They cannot be combined.
–15
5
ϭ
5x
5
– ALGEBRA–
340
A polynomial is the sum or difference of many terms and some have specific names:
8x
2
This is a monomial because there is one term.
3x + 2y This is a binomial because there are two terms.
4x
2
+ 2x – 6 This is a trinomial because there are three terms.

Laws of Exponents
■
When multiplying like bases, add the exponents: x
2
× x
3
= x
2 + 3
= x
5
■

When dividing like bases, subtract the exponents:
■
When raising a power to another power, multiply the exponents:
■
Remember that a fractional exponent means the root: ͙
ෆ
x = x
ᎏ
1
2
ᎏ
and ͙
3
x
ෆ
= x
ᎏ
1
3
ᎏ
The following is an example of a question involving exponents:
Solve for x:2
x + 2
= 8
3
.
a. 1
b. 3
c. 5
d. 7

e. 9
The correct answer is d. To solve this type of equation, each side must have the same base. Since 8 can
be expressed as 2
3
, then 8
3
= (2
3
)
3
= 2
9
. Both sides of the equation have a common base of 2, so set the expo-
nents equal to each other to solve for x. x + 2 = 9. So, x = 7.

Solving Linear Equations of One Variable
When solving this type of equation, it is important to remember two basic properties:
■
If a number is added to or subtracted from one side of an equation, it must be added to or subtracted
from the other side.
■
If a number is multiplied or divided on one side of an equation, it must also be multiplied or divided
on the other side.
1x
2
2
3
ϭ x
2ϫ3
ϭ x

6
x
5
x
2

= x
5 – 2
= x
3
– ALGEBRA–
341
Linear equations can be solved in four basic steps:
1. Remove parentheses by using distributive property.
2. Combine like terms on the same side of the equal sign.
3. Move the variables to one side of the equation.
4. Solve the one- or two-step equation that remains, remembering the two previous properties.
Examples
Solve for x in each of the following equations:
a. 3x – 5 = 10
Add 5 to both sides of the equation: 3x – 5 + 5 = 10 + 5
Divide both sides by 3:
x = 5
b. 3 (x – 1) + x = 1
Use distributive property to remove parentheses:
3x – 3 + x = 1
Combine like terms: 4x – 3 = 1
Add 3 to both sides of the equation: 4x – 3 + 3 = 1 + 3
Divide both sides by 4:
x = 1

c. 8x – 2 = 8 + 3x
Subtract 3x from both sides of the equation to move the variables to one side:
8x – 3x – 2 = 8 + 3x – 3x
Add 2 to both sides of the equation: 5x – 2 + 2 = 8 + 2
Divide both sides by 5:
x = 2

Solving Literal Equations
A literal equation is an equation that contains two or more variables. It may be in the form of a formula. You
may be asked to solve a literal equation for one variable in terms of the other variables. Use the same steps
that you used to solve linear equations.
5x
5
ϭ
10
5
4x
4
ϭ
4
4
3x
3
ϭ
15
3
– ALGEBRA–
342
Example
Solve for x in terms of a and b:2x + b = a

Subtract b from both sides of the equation: 2x + b – b = a – b
Divide both sides of the equation by 2:

Solving Inequalities
Solving inequalities is very similar to solving equations. The four symbols used when solving inequalities are
as follows:
■
Ͻ is less than
■
Ͼ is greater than
■
Յ is less than or equal to
■
Ն is greater than or equal to
When solving inequalities, there is one catch: If you are multiplying or dividing each side by a negative
number, you must reverse the direction of the inequality symbol. For example, solve the inequality
–3x + 6 Յ 18:
1. First subtract 6 from both sides:
2. Then divide both sides by –3:
3. The inequality symbol now changes:
Solving Compound Inequalities
A compound inequality is a combination of two inequalities. For example, take the compound inequality
–3 Ͻ x + 1 Ͻ 4. To solve this, subtract 1 from all parts of the inequality. –3 – 1 Ͻ x + 1 – 1 Ͻ 4 – 1. Simplify.
–4 Ͻ x Ͻ 3. Therefore, the solution set is all numbers between –4 and 3, not including –4 and 3.
x ՆϪ4
–3x
–3
Յ
12
–3

–3x ϩ 6 Ϫ 6 Յ 18 Ϫ 6
x ϭ
a Ϫ b
2
2x
2
ϭ
a Ϫ b
2
– ALGEBRA–
343

Multiplying and Factoring Polynomials
When multiplying by a monomial, use the distributive property to simplify.
Examples
Multiply each of the following:
1. (6x
3
)(5xy
2
) = 30x
4
y
2
(Remember that x = x
1
.)
2. 2x (x
2
– 3) = 2x

3
– 6x
3. x
3
(3x
2
+ 4x – 2) = 3x
5
+ 4x
4
– 2x
3
When multiplying two binomials, use an acronym called FOIL.
F Multiply the first terms in each set of parentheses.
O Multiply the outer terms in the parentheses.
I Multiply the inner terms in the parentheses.
L Multiply the last terms in the parentheses.
Examples
1. (x – 1)(x + 2) = x
2
+ 2x – 1x – 2 = x
2
+ x – 2
F O I L
2. (a – b)
2
= (a – b)(a – b) = a
2
– ab – ab – b
2

F O I L
Factoring Polynomials
Factoring polynomials is the reverse of multiplying them together.
Examples
Factor the following:
1. 2x
3
+ 2 = 2 (x
3
+ 1) Take out the common factor of 2.
2. x
2
– 9 = (x + 3)(x – 3) Factor the difference between two perfect squares.
3. 2x
2
+ 5x – 3 = (2x – 1)(x + 3) Factor using FOIL backwards.
4. 2x
2
– 50 = 2(x
2
– 25) = 2(x + 5)(x – 5) First take out the common factor and then factor the difference
between two squares.

Solving Quadratic Equations
An equation in the form y = ax
2
+ bx + c,where a, b, and c are real numbers, is a quadratic equation. In other
words, the greatest exponent on x is two.
– ALGEBRA–
344

Quadratic equations can be solved in two ways: factoring, if it is possible for that equation, or using the
quadratic formula.
By Factoring
In order to factor the quadratic equation, it first needs to be in standard form. This form is y = ax
2
+bx + c.
In most cases, the factors of the equations involve two numbers whose sum is b and product is c.
Examples
Solve for x in the following equation:
1. x
2
– 25 = 0
This equation is already in standard form. This equation is a special case; it is the difference
between two perfect squares. To factor this, find the square root of both terms.
The square root of the first term x
2
is x.
The square root of the second term 25 is 5.
Then two factors are x – 5 and x + 5.
The equation x
2
– 25 = 0
then becomes (x – 5)(x + 5) = 0
Set each factor equal to zero and solve x – 5 = 0 or x + 5 = 0
x = 5 or x = –5
The solution is {5, –5}.
2. x
2
+ 6x = –9
This equation needs to be put into standard form by adding 9 to both sides

of the equation. x
2
+ 6x + 9 = –9 + 9
x
2
+ 6x + 9 = 0
The factors of this trinomial will be two numbers whose sum is 6 and whose product is 9. The fac-
tors are x + 3 and x + 3 because 3 + 3 = 6 and 3 × 3 = 9.
The equation becomes (x + 3)(x + 3) = 0
Set each factor equal to zero and solve x + 3 = 0 or x + 3 = 0
x = –3 or x = –3
Because both factors were the same, this was a perfect square trinomial. The solution is {–3}.
3. x
2
= 12 + x
This equation needs to be put into standard form by subtracting 12 and x from both sides of the
equation. x
2
– x – 12 = 12 – 12 + x – x
x
2
– x – 12 = 0
– ALGEBRA–
345
Since the sum of 3 and –4 is –1, and their product is –12, the equation factors to (x + 3)
(x – 4) = 0
Set each factor equal to zero and solve: x + 3 = 0 or x – 4 = 0
x = –3 or x = 4
The solution is {–3, 4}.
By Quadratic Formula

Solving by using the quadratic formula will work for any quadratic equation, especially those that are not fac-
torable.
Solve for x:
x
2
+ 4x = 1
Put the equation in standard form. x
2
+ 4x – 1 = 0
Since this equation is not factorable, use the quadratic formula by identifying the value of a, b, and
c and then substituting it into the formula. For this particular equation, a = 1, b = 4, and c = –1.
The solution is .
The following is an example of a word problem incorporating quadratic equations:
A rectangular pool has a width of 25 feet and a length of 30 feet. A deck with a uniform width sur-
rounds it. If the area of the deck and the pool together is 1,254 square feet, what is the width of
the deck?
ͭ –2 ϩ 2 5
, –2 – 2 5 ͮ
x ϭ –2
; 2 5
x ϭ
–4
2
;
22 5
2
x ϭ
–4
; 2 20
2

x ϭ
–4
; 2 16 ϩ 4
2
x ϭ
–4
; 2 4
2
Ϫ 41121–12
2112
x ϭ
–b
; 2 b
2
–4ac
2a
– ALGEBRA–
346
Begin by drawing a picture of the situation. The picture could be similar to the following
figure.
Since you know the area of the entire figure, write an equation that uses this information. Since we are
trying to find the width of the deck, let x = the width of the deck. Therefore, x + x + 25 or 2x + 25 is the width
of the entire figure. In the same way, x + x + 30 or 2x + 30 is the length of the entire figure.
The area of a rectangle is length × width, so use A = l × w.
Substitute into the equation: 1,254 = (2x + 30)(2x + 25)
Multiply using FOIL: 1,254 = 4x
2
+ 50x + 60x + 750
Combine like terms: 1,254 = 4x
2

+ 110x + 750
Subtract 1,254 from both sides: 1,254 – 1,254 = 4x
2
+ 110x + 750 – 1,254
0 = 4x
2
+ 110x – 504
Divide each term by 2: 0 = 2x
2
+55x – 252
Factor the trinomial: 0 = (2x + 63 )(x – 4)
Set each factor equal to 0 and solve 2x + 63 = 0 or x – 4 = 0
2x = –63 x = 4
x = –31.5
Since we are solving for a length, the solution of –31.5 must be rejected. The width of the deck is 4 feet.

Rational Expressions and Equations
Rational expressions and equations involve fractions. Since dividing by zero is undefined, it is important to
know when an expression is undefined.
The fraction is undefined when the denominator x – 1 = 0; therefore, x =1.
5
x Ϫ 1
x
x
x
x
25
30
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347

You may be asked to perform various operations on rational expressions. See the following examples.
Examples
1. Simplify .
2. Simplify .
3. Multiply .
4. Divide .
5. Add .
6. Subtract .
7. Solve .
8. Solve .
Answers
1.
2.
3.
4.
5.
6.
3x ϩ 18 Ϫ x ϩ 2
3x
ϭ
2x ϩ 20
3x
1 ϩ 3x
xy
a1a ϩ 22
1a ϩ 1 21a ϩ 22
ϫ
21a ϩ 12
a1a Ϫ 32
ϭ

2
a Ϫ 3
4x1x ϩ 42
2x
2
1x Ϫ 421x ϩ 42
ϭ
2
x1x Ϫ 42
1x ϩ 3 21x Ϫ 32
31x Ϫ 32
ϭ
1x ϩ 3 2
3
x
2
b
x
3
b
2
ϭ
1
xb
1
x
ϭ
1
4
ϩ

1
6
2
3
x ϩ
1
6
x ϭ
1
4
x ϩ 6
x
–
x –2
3x
1
xy
ϩ
3
y
a
2
ϩ 2a
a
2
ϩ 3a ϩ 2
Ϭ
a
2
–3a

2a ϩ 2
4x
x
2
–16
ϫ
x ϩ 4
2x
2
x
2
Ϫ 9
3x Ϫ 9
x
2
b
x
3
b
2
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348