Week 3 Day 1 (Beginning of Algebra)

Algebra Review

The rules for algebra are basically the same as the rules for arithmetic, except now we include letters used as variables.

Understood Multiplication p. 85

When a variable is next to another variable or next to a number, this is a convention in algebra that means these two are multiplied together. Ex.: 4a means “4 times a,” and xyz means “x times y times z.”

Letters to Avoid as Variables

Some letters do not make good variables because in handwriting they can look very similar to numerals. These include t, o, z, s, and l. Depending on your handwriting, the lowercase t looks very similar to +, both lowercase and capital o/O look like 0, z/Z looks like 2, s/S looks like 5, and both lowercase l and capital I can look like 1. You also want to avoid i and e because these lowercase letters have special meanings and values.

Basic Set Theory

Terminology

Set: groups of objects or numbers, e.g. {1, 2, 3, …}; distinct from lists, e.g. 3, 1, 2

Element: a member of a set, e.g. 3 Î {1, 2, 3, …}

Subset: a set within a set, e.g. {2, 3} Ì {1, 2, 3, …}

Universal set: the set of all sets, i.e. U

Empty set (or null set): a set with no elements, i.e., Æ, {}, or {Æ}

Rules: methods of describing sets by describing the properties of their elements, e.g.,

{"x | x > 3, x is a whole number}

            (read: For all x such that x is greater than three and x is a whole number)

Rosters: methods of describing sets by their listing their members, e.g., {4, 5, 6, …}

Venn Diagrams (aka Euler Circles): pictorially describe sets

Finite sets: sets that are countable and come to a stop, e.g. {1, 2, 3, 4}

Infinite sets: sets that are uncountable and continue forever, e.g. {…2, 3, 4, …} or {1, 2, 3, …}

Equal sets: sets that have the exact same members, e.g. {1, 2, 3} = {3, 2, 1}

Equivalent sets: sets that have the same number of members: e.g. {1, 2, 3} ~ {a, b, c}

Unions: the set including the members of two sets, e.g. {1, 2} È {3, 4} = {1, 2, 3, 4}

Intersections: the set including only the overlapping members of two sets,

e.g. {1, 2} Ç {2, 3} = {2}      OR       {1, 2} Ç {3, 4} = Æ ß Use the null set when there are no overlapping members

Variables & Algebraic Expressions p. 86-7

Variables: letters denoting elements; usually a letter stands for a specific number or group of numbers

Variables change verbal expressions into algebraic expressions. I want to emphasize the turnaround words and other keywords are important for making this translation.

Evaluating expressions (p. 88) just means solving those expressions: we translate from the written or spoken word into numerals (0, 1, 2, etc.), operators (+, –, ´, ¸) grouping symbols ({[()]}), and variables (x, y, a, b); “plug in” the values given (e.g., x = 4); and finally do the arithmetic to solve.

Equations

Mathematical expressions set equal to each other are equations. To solve equations, we ALWAYS must do the same thing to both sides of the equation to maintain the balance or equality. We must also perform opposite operations from those stated in the equation, and we must isolate the variable on one side of the equal sign in order to find its value. See ex. on p. 90.

The final step in solving any equation is to check your answer. Do this by plugging it in for the resolved variable and seeing whether or not the equality stands. See ex. on p. 91-92.

Solving Functions p. 92

f(x) = algebraic expression <-- This is a function.

"F of x" is the function of x.

START HERE WEDNESDAY

Literal Equations p. 94

These equations are called “literal” because they have only letters, no numbers. We solve these by isolating the desired letter alone on one side of the equal sign. See ex. 1-5 on p. 130-1.

Averages, Ratios, Proportions

Ratios & Proportions

Ratios are a method of comparing two or more variables, usually represented in one of several ways:          a:b       a/b       ‘a is to b’

Note that the first or top letter is stated first while the second or bottom letter is stated last.

Proportions are a relationship between two ratios set equal to each other:

We would read these as ‘p is to q as x is to y.’ Note that the first letters in each ‘is to’ relationship are the numerators, the second letters are the denominators, and ‘as’ is translated as an equal sign.

These can be solved either as literal equations, as in our first set of practice problems, or for value, as in our second set of practice problems. Either way, the process is the same: undoing the division by multiplying both sides by one of the variables in an attempt to isolate the desired variable.

Systems of Equations

We have two methods to solve systems or groups of equations with the same variables: elimination and substitution. It is important to note for both the GRE and the GMAT that a system of equations can only be solved when the number of equations is equal to the total number of unique variables. In other words, if your variables are x and y, you need two equations in order to solve the system; if your variables are x, y, and z, you need three equations in order to solve the system.

Method 1: Elimination

1. If the variables in both equations all have different coefficients (numbers in front of the variables), multiply one equation by a number to make the coefficient for the chosen variable in that equation the same as the coefficient for the paired variable in the other equation (it must be the same number applied to each part of the equation on the left and the right sides of the equal sign).

2. Add or subtract the two equations to make the chosen variable cancel out.

3. Solve for the remaining variable.

4. Insert that value into one of the original equations to solve for the value of the variable that was canceled out in step 2.

Example:

Method 2: Substitution

1. Solve one equation for one variable by isolating the chosen variable on one side.

2. Insert that solution into the OTHER equation and solve for the OTHER variable.

3. Insert that solution into the original equation to solve for the original variable.

Monomials and Polynomials

Monomials are expressions consisting of one term. Examples: 9x, 4a², and 3mpxz²

Polynomials consist of more than one term. Examples: x + y, y² – x², and x² + 3x + 5y²

            Binomials have exactly two terms. Example: x + y

            Trinomials have exactly three terms. Example: y² + 9y + 8

The coefficient is the number in front of the variable. Example: in 9x, 9 is the coefficient.

Polynomials can be arranged in ascending or descending order.

Ascending order examples: x + x² + x³ or 5x + 2x² – 3x³ + x⁵  ß Typically preferred order

Descending order examples: x³ + x² + x or 2x⁴ + 3x² + 7x

Operations with Monomials and Polynomials

The general rule is that you must combine like with like in addition and subtraction, but also note that the changes occur with the coefficients and not the variables, and that the rules for signed numbers apply as usual.

When multiplying, you multiply the coefficients, and add the exponents on the variables that are the same – and don’t forget the invisible exponent of 1 on variables that have no visible exponent. However, if the exponent is on the parentheses, you will multiply the interior exponent(s) by the exterior exponent.

When dividing, you divide the coefficients, and subtract the exponents on the variables that are the same – again, don’t forget the invisible 1. In some cases, you can simply cancel out the like terms.