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 Algebra 21... The follow
Trang 2Tr a n s l a t i n g E x p r e s s i o n s a n d E q u a t i o n s
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
Algebra
21
Trang 3The 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:
Fifteen less than five times the number equals the product of 10 and x.
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.
C o m b i n i n g L i k e Te r m s a n d P o l y n o m i a l s
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 coefficoeffi-cient 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.
–2x3y 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.
4x2y + –10x2y These terms are also like terms, and the sum is –6x2y.
2xy2+ 9x2y These terms are not like terms because the variables, taken with their powers,
are not exactly the same They cannot be combined
–15
5
– A L G E B R A –
Trang 4A polynomial is the sum or difference of many terms and some have specific names:
8x2 This is a monomial because there is one term.
3x + 2y This is a binomial because there are two terms.
4x2+ 2x – 6 This is a trinomial because there are three terms.
L a w s o f E x p o n e n t s
■ When multiplying like bases, add the exponents: x2× x3= x2 + 3 = x5
■ 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:x = x12and 3
x
= x13
The following is an example of a question involving exponents:
Solve for x: 2 x + 2= 83
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 23, then 83= (23)3= 29 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.
S o l v i n g L i n e a r E q u a t i o n s o f O n e Va r i a b l e
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
1x223 x2 3 x6
x5
Trang 5Linear 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
S o l v i n g L i t e r a l E q u a t i o n s
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
4x
4
3x
3
– A L G E B R A –
Trang 6Solve 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:
S o l v i n g I n e q u a l i t i e s
Solving inequalities is very similar to solving equations The four symbols used when solving inequalities are
as follows:
■ is less than
■ is greater than
■ is less than or equal to
■ 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 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 x + 1 4 To solve this, subtract 1 from all parts of the inequality –3 – 1 x + 1 – 1 4 – 1 Simplify –4 x 3 Therefore, the solution set is all numbers between –4 and 3, not including –4 and 3
x
–3x
–3
–3x
xa
2
2x
2
...a 3x – = 10
Add to both sides of the equation: 3x – + = 10 + 5
Divide both sides by 3:
x = 5
b (x – 1) + x = 1< /b>
Use...
3x – + x = 1< /i>
Combine like terms: 4x – = 1< /i>
Add to both sides of the equation: 4x – + = + 3
Divide both sides by 4:
x = 1< /i>
c... direction of the inequality symbol For example, solve the inequality
–3x + 6 18 :
1 First subtract from both sides:
2 Then divide both sides by –3:
3