GRE Math Definitions, Formulas and Problems (Algebra)

AL 3a. What is the value of f(x) = 2x^2 - 7x + 23, when x = -2?

2(-2)^2 - 7(-2) + 23 = 8 + 14 + 23 = 45

AL 8b. 25x + 16 >= 10 - x

26x > -6 x = -6/26 x >= -6/26 or -3/13

AL 8c. 16 + x > 8x - 12

28 > 7x 7x < 28 x < 4

AL 1b. Three times x is squared and the result is divided by 7

((3x)^2)/7

AL 1c. The product of x + 4 and y is added to 18.

(x+4)(y) + 18

AL 5g. (x^10)(y-1)/(x^-5)(y^5)

(x^10)(x^5)/(y^5)(y^1) = x^15/y^6

AL 8a. -3x > 7 + x

-4x > 7 x = 7/-4 x < -1.75

AL 3c. What is the value of f(x) = (5/3)x - 7 when x = 0? (revise and solve)

0 - 7 = -7

AL 1a. The square of y subtracted from 5 and the result is multiplied by 37. (revise)

37(5 - y^2)

AL 2a. 3x^2 - 6 + x + 11 - x^2 + 5x (simplify)

3x^2 - x^2 - 6 + 11 - x^2 + 5x = 14x + 5 - x^2

AL 6c. 5(x + 2) = 1 - 3x

5x + 10 = 1 - 3x, 8x = -9 so x = -9/8

Rise Over Run

Ratio of Rise(y2-y1) over Run(x2-x1)

Degree (Math)

The total variable in a expression

Exponent Rule 3

When Divided, exponents are subtraced

Functions

an algebric expression that pairs input and output to create a function i.e. input -> f(x) = 3x + 5 <- output

AL 5f. (5^0)(d^3)

d^3

Exponent Rule 7

exponents can multiply by another exponent. i.e. (2^5)^2 = 2^10

AL 5c. r^12/r^4 simplify

r^8

Coordinate Geometry

the use of algebra to study geometric properties

AL 6d. (x + 6)(2x - 1) = 0

x = - 6 or .5

AL 7b. 3x - y = -5 x + 2y = 3

x = -2y + 3 3(-2y + 3) -y = -5 -6y + 9 -y = -5 -7y = -14 y = 2 x = -1

Quadratic Formula

x = -b ± √(b² - 4ac)/2a

*AL 6f. x^2 - x - 1 = 0

x^2 - x = 1 x = 1 + √5/2 or 1 - √5/2

Straight Line Equation

y = sx + yi i.e. line = slope(x) + y intercept

Slope Equation

y2-y1/x2-x1

AL 5b. (s^7)(t^7) simplify

(st)^7

Circle Equation

(x-h)^2 + (y-k)^2 = r^2

Exponent Rule 6

(x/y)^2 = (x^2/y^2) i.e. (3/4) = 3^2/4^2

Exponent Rule 5

(xy)^a = (x)^a x (y)^a and vice versa i.e. (2^3)(3^3) = 6^3

AL 6b. 12 - 5x = x + 30

-18 = 6x so x = -3

AL 4b. What is the value of y/[y] when g(y) = -2.

-2/[-2] = -1

Equilvalent Equation Rules

1. When add or subtracted from both sides 2. When multiplied or dividend using no zero the equation is still equivalent

AL 5e. (w^5)^-3

1/w^15

AL 2b. 3(5x - 1) - x + 4 (simplify)

15x - 3 - x + 4 = 14x + 1

*AL 7c. 15x - 18 - 2y = -3x + y 10x + 7y + 20 = 4x +2

18x = 18 + 3y x = 1 + 3y/18 or x - 1 = 3y 18x -3y = 18 6x + 7y = -18 12x -10y = 36 x = .5 y = -3

AL 4a. What is the value of y/[y] when g(y) = 2.

2/[2] = 1

AL 4c. What is the value of y/[y] when g(y) = 2-(-2).

2/[2]=1 -2/[-2]=-1 1-(-1) = 2

AL 14. Two cars started from the same point and traveled on a straight course in opposite directions for 2 hours, at which time they were 208 miles apart. If one car traveled, on average, 8 miles per hour faster than the other car, what was the average speed of each car for the 2-hour trip?

208 = 2a - 2b with a range of 4 48 56 48 56 car 1 = 56 car 2 = 48

AL 3b. What is the value of f(x) = x^3 - 2x^2 + x - 2, when x = 2?

2^3 - 2(2)^2 + 2 - 2 = 8 - 8 + 2 - 2 = 0

AL 10. If the ratio of 2x to 5y is 3 to 4, what is the ratio of x to y?

2x/5y = 3/4 2x/5y = 90/120 x = 45 y = 24 x = 15 y = 8

*AL 13. Pat invested a total of $3,000. Part of the money was invested in a money market account that paid 10 percent simple annual interest, and the remainder of the money was invested in a find that paid 8 percent simple annual interest. If the total interest earned at the end of the first year from these investments was $256, how much did Pat invest at 10 percent and how much at 8 percent?

3256 = 3000 + .1m + .08r 256 = .1m + .08r .1m = 256 - .08r 1m + .8r = 2560 800 = 800 2200 = 1760 m = 800 r = 2200

*AL 5d. (2a/b)^5 simplify

32a^5(b^-5) or 32a^5/b^5

AL 15. A group can charter a particular aircraft at a fixed total cost. If 36 people charter the aircraft rather than 40 people, then the cost per person is greater by $12. (a) What is the fixed total cost to charter the aircraft? (b) What is the cost per person if 40 people charter the aircraft?

36 x a(p + 12) = 40 x a(p - 12) 36(p + 12) = 40(p - 12) 36p + 432 = 40p - 480 4p = 912 p = 228/2 = 114 f = 8640/2 = 4320 a = 216/2 = 108

*AL 5h. (3x/y)^2/(1/y)^5

3x^2(y^-2)/(y^-5) = 9x^2(y^3)

AL 12. A theater sells children's tickets for half the adult ticket price. If 5 adult tickets and 8 children's tickets cost a total of $81, what is the cost of an adult ticket?

5a + 8c = 81 c = a(1/2) 5a + 8 x a x 1/2 = 81 9a = 81 a = 9 b = 4.5

*AL 6e. x^2 + 5x - 14 = 0

5x + x^2 = 14 x = 2 and -7

AL 6a. 5x -7 = 28

5x = 35 so x = 7

AL 2d. (2x + 5)(3x - 1) (simplify)

6x^2 - 2x + 15x - 5 = 6x^2 +13x - 5

Algebra

A branch of mathematics that involves expressions, variables, equations, inequalities and functions.

Algebraic Expression

A combination of variables, numbers, and at least one operation.

Coefficient

A number multiplied by a variable. i.e. 2x

Ordered Pair

A pair of numbers that can be used to locate a point on a coordinate plane

Constant Term

A term without a variable i.e. 1

Compound Interest

A=P(1+r/n)^nt

Inequalities

Algebraic statements that have ≠, <, >, ≤, or ≥ as their symbols of comparison.

Linear Equation

An equation whose graph is a line.

Polynomial

An expression with 1 or more terms i.e. 2x + 1

Exponent Rule 4

Exponents to the 0 is 1 unless the coefficent is 0

Exponent Rule 2

Exponets with similar variables and terms will add together when the are multiplied i.e. (3^2)(3^1) = 3^3

Quadrants

Four regions into which a coordinate plane is divided by the x-axis and the y-axis

Can Negative Numbers be Squares?

No

Can you Divide by 0?

No

Simple Interest

Principal x Rate x Time

Solving Two Variables Linear Equations

Subsitute equation to eliminate for y and solve for x and vice versa

Parabola

The U-shaped graph of a quadratic function

Degree of a polynomial

The greatest degree of any term in the polynomial

Domain

The set of input values of a function. i.e. f(x) = x^2-4 domain is anything between -2 and 2

Applications

Translating a verbal description into a algebraic expression

Equivalent Equation

Two or more equations with the same solution. i.e. x + 1 = 2x + 2

Sovling Quadratic Equation

Using Quadratic Formula or Factoring

Exponent Rule 1

a negative exponent means the value will be a decimal i.e. 4^-3 = 1/64

Term

a number or variable i.e. 2x

*AL 18. In the xy-plane, find the following. (a) The slope and y-intercept of the line with equation 2y + x = 6 (b) The equation of the line passing through the point (3, 2) with y-intercept 1 (c) The y-intercept of a line with slope 3 that passes through the point (-2, 1) (d) The x-intercepts of the graphs in parts (a), (b), and (c)

a. x = 0, 2, 4 , 6 y = 3, 2, 1, 0 (0,3)(2,2)(4,1)(6,0) y2 - y1/x2 - x1 2 - 3/2-0 = -1/2 b. y = s(x) + 1 2 = s(3) + 1 s = 1/3 y = x/3 + 1 X c. 1 = 3(1) + yi yi = -2 X d. 6, -3, and -7/3

Pythagorean Theorem

a²+b²=c²

*AL 16. An antiques dealer bought c antique chairs for a total of x dollars. The dealer sold each chair for y dollars. (a) Write an algebraic expression for the profit, P, earned from buying and selling the chairs. (b) Write an algebraic expression for the profit per chair.

c(x) - c(y) = p p/c = profit per chair

Linear Inequalities

contains a variable and an inequality sign such as < or >

Quadratic Equation

is an equation that can be written in the form ax^2 + bx + c = 0, where a is not zero.

Variable

letter that represents a quantity whose value is unknown

AL 5a. (n^5)(n^-3) simplify

n^2

AL 11. Kathleen's weekly salary was increased by 8 percent to $712.20. 'What was her weekly salary before the increase?

s(1) + s(.08) = 712.20 1s + .08s = 712.20 1.08s = 712.20 1.08s/1.08 = 712.20/1.08 s = 659.44

Factoring

spliting equations

Like Terms

terms that have the same variables with the exact same exponents

AL 9. For a given two-digit positive integer, the tens digit is 5 more than the units digit. The sum of the digits is 11. Find the integer.

x + (y+5) = 11

AL 2c. (x^2 - 16)/(x-4), and x is not 4 (simplify)

x + 4

AL 7a. x + y = 24 x - y = 18

x = 24 - y 24 - y - y = 18 24 - 2y = 18 - 2y = -6 y = 3 x = 21