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