Example: 10 Ͼ 5 so (10)(–3) Ͻ (5)(–3) –30 Ͻ –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 Ͼ 21, find x. ■ Isolate the variable: 7 – 2x Ͼ 21 –7 –7 –2x Ͼ 14 ■ Because you are dividing by a negative number, the inequality symbol changes direction: ᎏ – – 2 2 x ᎏ Ͼ ᎏ – 14 2 ᎏ x Ͻ –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 Ͻ ax + b Ͻ d, isolate the letter by performing the same operation on each member of the equation. Example: If –10 Ͻ –5y – 5 Ͻ 15, find y. ■ Add five to each member of the inequality: –10 + 5 Ͻ –5y – 5 + 5 Ͻ 15 + 5 –5 Ͻ –5y Ͻ 20 ■ Divide each term by –5, changing the direction of both inequality symbols: ᎏ – – 5 5 ᎏ Ͻ ᎏ – – 5 5 y ᎏ Ͻ ᎏ – 20 5 ᎏ = 1 Ͼ y Ͼ –4 ■ The solution consists of all real numbers less than 1 and greater than –4. – THE GRE QUANTITATIVE SECTION– 175 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 ϫ 6 Ͼ 2 When 14 is added to a number x, the sum is less than 21. x ϩ 14 Ͻ 21 The sum of a number x and four is at least nine. x ϩ 4 Ն 9 When seven is subtracted from a number x, the difference is at most four. x – 7 Յ 4 – THE GRE QUANTITATIVE SECTION– 176 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 ᎏ x 3 – x 1 ᎏ , the input x = 2 yields an output of ᎏ 2 3( – 2) 1 ᎏ = ᎏ 6 1 ᎏ or 6. In function notation, the expression ᎏ x 3 – x 1 ᎏ is deemed a function and is indicated by a letter, usually the letter f: f (x) = ᎏ x 3 – x 1 ᎏ It is said that the expression ᎏ x 3 – x 1 ᎏ defines the function f (x). For this example with input x = 2 and out- put 6, you write f(2) = 6. The output 6 is called the value of the function with an input x = 2. The value of the same function corresponding to x = 4 is 4, since ᎏ 4 3( – 4) 1 ᎏ = ᎏ 1 3 2 ᎏ = 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) = x 2 + ͙x ෆ + 10, f (x) is undefined for x = –10, since one of the terms would be ͙–10 ෆ . – THE GRE QUANTITATIVE SECTION– 177 Geometry Review 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. PARALLEL AND PERPENDICULAR LINES 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. l 1 l 2 l l 2 1 A B l – THE GRE QUANTITATIVE SECTION– 178 Angles An angle is formed by an endpoint, or vertex, and two rays. NAMING ANGLES 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: ЄA. 2. An angle can be represented by a number written in the interior of the angle near the vertex: Є1. 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 ЄBAD or as ЄDAB; Є2 can be written as ЄDAC or as ЄCAD. CLASSIFYING ANGLES 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 AC D B Endpoint, or Vertex ray ray – THE GRE QUANTITATIVE SECTION– 179 ■ 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. COMPLEMENTARY ANGLES Two angles are complimentary if the sum of their measures is equal to 90 degrees. 1 2 ∠1 + m∠2 = 90° Complementary Angles m Straight Angle 180° Obtuse Angle Right Angle Symbol – THE GRE QUANTITATIVE SECTION– 180 . = –10, since one of the terms would be ͙–10 ෆ . – THE GRE QUANTITATIVE SECTION 177 Geometry Review About one-third of the questions on the Quantitative section of the GRE have to do with geometry complimentary if the sum of their measures is equal to 90 degrees. 1 2 ∠1 + m∠2 = 90° Complementary Angles m Straight Angle 180 ° Obtuse Angle Right Angle Symbol – THE GRE QUANTITATIVE SECTION 180 . angle. l 1 l 2 l l 2 1 A B l – THE GRE QUANTITATIVE SECTION 1 78 Angles An angle is formed by an endpoint, or vertex, and two rays. NAMING ANGLES There are three ways to name an angle. 1. An angle can be named by the vertex