The gre anlytycal writting section 8 ppt

Perpendicular lines intersect at a 90-degree angle... An angle can be named by the vertex when no other angles share the same vertex:A.. An angle can be represented by a number written i

–30 –15

Solving Linear Inequalities

To solve a linear inequality, isolate the letter and solve the same as you would in a first-degree equation Remember to reverse the direction of the inequality sign if you divide or multiply both sides of the equation

by a negative number

Example:

If 7 – 2x  21, find x.

■ Isolate the variable:

7 – 2x 21

–7 –7

–2x 14

■ Because you are dividing by a negative number, the inequality symbol changes direction:

–

–

2

2

x



–

14

2



x –7

■ The answer consists of all real numbers less than –7

Solving Combined (or Compound) Inequalities

To solve an inequality that has the form c  ax + b  d, isolate the letter by performing the same operation

on each member of the equation

Example:

If –10 –5y – 5  15, find y.

■ Add five to each member of the inequality:

–10 + 5 –5y – 5 + 5  15 + 5

–5 –5y  20

■ Divide each term by –5, changing the direction of both inequality symbols:

–

–

5

5

–

–

5

5

y

 –

20 5

= 1 y  –4

■ The solution consists of all real numbers less than 1 and greater than – 4

Translating Words into Numbers

The most important skill needed for word problems is being able to translate words into mathematical oper-ations The following list will give you some common examples of English phrases and their mathematical equivalents

■ “Increase” means add

Example:

A number increased by five = x + 5.

■ “Less than” means subtract

Example:

10 less than a number = x – 10.

■ “Times” or “product” means multiply

Example:

Three times a number = 3x.

■ “Times the sum” means to multiply a number by a quantity

Example:

Five times the sum of a number and three = 5(x + 3).

■ Two variables are sometimes used together

Example:

A number y exceeds five times a number x by ten.

y = 5x + 10

■ Inequality signs are used for “at least” and “at most,” as well as “less than” and “more than.”

Examples:

The product of x and 6 is greater than 2.

x 6  2

When 14 is added to a number x, the sum is less than 21.

x 14  21

The sum of a number x and four is at least nine.

x 4  9

When seven is subtracted from a number x, the difference is at most four.

Assigning Variables in Word Problems

It may be necessary to create and assign variables in a word problem To do this, first identify an unknown and a known You may not actually know the exact value of the “known,” but you will know at least some-thing about its value

Examples:

■ Max is three years older than Ricky

Unknown = Ricky’s age = x.

Known = Max’s age is three years older

Therefore, Ricky’s age = x and Max’s age = x + 3.

■ Siobhan made twice as many cookies as Rebecca

Unknown = number of cookies Rebecca made = x.

Known = number of cookies Siobhan made = 2x.

■ Cordelia has five more than three times the number of books that Becky has

Unknown = the number of books Becky has = x.

Known = the number of books Cordelia has = 3x + 5.

Algebraic Functions

Another way to think of algebraic expressions is to think of them as “machines” or functions Just like you

would a machine, you can input material into an equation that expels a finished product, an output or

solu-tion In an equation, the input is a value of a variable x For example, in the expression x3–x1, the input

x = 2 yields an output of23(–2)1= 61or 6 In function notation, the expression x3–x1is deemed a function and is

indicated by a letter, usually the letter f:

f (x) = x3–x1

It is said that the expression x3–x1defines the function f (x) For this example with input x = 2 and out-put 6, you write f(2) = 6 The outout-put 6 is called the value of the function with an inout-put x = 2 The value of the same function corresponding to x = 4 is 4, since 43(–4)1= 132= 4

Furthermore, any real number x can be used as an input value for the function f(x), except for x = 1, as this substitution results in a 0 denominator Thus, it is said that f(x) is undefined for x = 1 Also, keep in mind

that when you encounter an input value that yields the square root of a negative number, it is not defined under the set of real numbers It is not possible to square two numbers to get a negative number For

exam-ple, in the function f (x) = x2+ x + 10, f (x) is undefined for x = –10, since one of the terms would be –10

 G e o m e t r y R e v i e w

About one-third of the questions on the Quantitative section of the GRE have to do with geometry How-ever, you will only need to know a small number of facts to master these questions The geometrical concepts tested on the GRE are far fewer than those that would be tested in a high school geometry class Fortunately,

it will not be necessary for you to be familiar with those dreaded geometric proofs! All you will need to know

to do well on the geometry questions is contained within this section

Lines

The line is a basic building block of geometry A line is understood to be straight and infinitely long In the following figure, A and B are points on line l.

The portion of the line from A to B is called a line segment, with A and B as the endpoints, meaning that

a line segment is finite in length

P ARALLEL AND P ERPENDICULAR L INES

Parallel lines have equal slopes Slope will be explained later in this section, so for now, simply know that

par-allel lines are lines that never intersect even though they continue in both directions forever

Perpendicular lines intersect at a 90-degree angle.

l1

l2

l l

2

1

An angle is formed by an endpoint, or vertex, and two rays.

N AMING A NGLES

There are three ways to name an angle

1 An angle can be named by the vertex when no other angles share the same vertex:A.

2 An angle can be represented by a number written in the interior of the angle near the vertex:1

3 When more than one angle has the same vertex, three letters are used, with the vertex always being the

middle letter: 1 can be written as BAD or as DAB; 2 can be written as DAC or as CAD

C LASSIFYING A NGLES

Angles can be classified into the following categories: acute, right, obtuse, and straight

■ An acute angle is an angle that measures less than 90 degrees.

Acute Angle

1 2

D B

Endpoint, or Vertex

ray

ray

■ A right angle is an angle that measures exactly 90 degrees A right angle is sumbolized by a square at the

vertex

■ An obtuse angle is an angle that measures more than 90 degrees, but less than 180 degrees.

■ A straight angle is an angle that measures 180 degrees Thus, both its sides form a line.

C OMPLEMENTARY A NGLES

Two angles are complimentary if the sum of their measures is equal to 90 degrees.

1 2

Complementary Angles

Straight Angle

180°

Obtuse Angle

Right Angle

Symbol