CLEP College Algebra – Algebra Principles

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then a < c

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if a < b and b < c

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then a + c < b + d

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if a < b and c < d

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then ac < bc

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if a 0

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then bc < ac

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if a < b and c < 0

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The simplest equations to solve

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are linear equations that have only one variable. They contain only constant numbers and a single variable without an exponent. For example:

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The central technique to linear equations

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is to add, subtract, multiply, or divide both sides of the equation by the same number in order to isolate the variable on one side of the equation. Once the variable is isolated, the other side of the equation is the value of the variable.

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Quadratic equations

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can be expressed in the form ax^2 + bx + c = 0, where a is not zero (if it were zero, then the equation would not be quadratic but linear).

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Quadratic equations can also be solved

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using factorization (the reverse process of which is expansion, but for two linear terms is sometimes denoted foiling).

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All quadratic equations

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will have two solutions in the complex number system, but need not have any in the real number system.

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exponential equation

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is an equation of the form aX = b for a > 0, which has solution

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when b > 0

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Elementary algebraic techniques are used to rewrite a given equation in the above way before arriving at the solution. then, by subtracting 1 from both sides of the equation, and then dividing both sides by 3 we obtain

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logarithmic equation

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is an equation of the form log`a^X = b for a > 0, which has solution

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radical equation

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is an equation of the form X^m/n = a, for m, n integers, which has solution

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system of linear equations

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two equations in two variables, it is often possible to find the solutions of both variables that satisfy both equations.

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Elimination method

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An example of solving a system of linear equations is by using the elimination method: Multiplying the terms in the second equation by 2: Adding the two equations together to get: which simplifies to Since the fact that x = 2 is known, it is then possible to deduce that y = 3 by either of the original two equations (by using 2 instead of x) The full solution to this problem is then Note that this is not the only way to solve this specific system; y could have been solved before x.

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substitution

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An equivalent for y can be deduced by using one of the two equations. Using the second equation: Subtracting 2x from each side of the equation: and multiplying by -1: Using this y value in the first equation in the original system: Adding 2 on each side of the equation: which simplifies to Using this value in one of the equations, the same solution as in the previous method is obtained. Note that this is not the only way to solve this specific system; in this case as well, y could have been solved before x.

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an operation

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is an action or procedure which produces a new value from one or more input values.

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unary and binary

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There are two common types of operations:

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Unary operations

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involve only one value, such as negation and trigonometric functions.

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Binary operations

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take two values, and include addition, subtraction, multiplication, division, and exponentiation.

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The logical values true and false

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can be combined using logic operations, such as and, or, and not.

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Vectors

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can be added and subtracted.

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Rotations

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can be combined using the function composition operation, performing the first rotation and then the second.

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Operations on sets

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include the binary operations union and intersection and the unary operation of complementation.

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Operations on functions

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include composition and convolution

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Operations

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may not be defined for every possible value.

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domain

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The values for which an operation is defined form a set called its

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range

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The set which contains the values produced is called the codomain, but the set of actual values attained by the operation is its

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nonnegative numbers

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the squaring operation only produces

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nonnegative numbers

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the codomain is the set of real numbers but the range is the

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Operations can involve dissimilar objects

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A vector can be multiplied by a scalar to form another vector

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scalar

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the inner product operation on two vectors produces a

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operands, arguments, or inputs

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The values combined are called

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value, result, or output

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the value produced is called

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two inputs

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Operations can have fewer or more than

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An operation ω

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is a function of the form ω : V → Y, where V ⊂ X1 × ... × Xk.

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The sets Xk

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are called the domains of the operation

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the set Y

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is called the codomain of the operation

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the fixed non-negative integer k (the number of arguments)

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is called the type or arity of the operation

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has arity one

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a unary operation

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has arity two

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a binary operation

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nullary operation

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An operation of arity zero is simply an element of the codomain Y, called a

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k-ary operation

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An operation of arity k is called a

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(k+1)-ary relation that is functional on its first k domains

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k-ary operation is a

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finitary operation

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referring to the finite number of arguments (the value k)

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operation

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implies that the domain of the function is a power of the codomain (i.e. the Cartesian product of one or more copies of the codomain)

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operation

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is synonymous with function, map and mapping, that is, a relation, for which each element of the domain (input set) is associated with exactly one element of the codomain (set of possible outputs).

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equation

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a mathematical statement that asserts the equality of two expressions, this is written by placing the expressions on either side of an equals sign (=).

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Equations

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often express relationships between given quantities, the knowns, and quantities yet to be determined, the unknowns.

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unknowns

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are denoted by letters at the end of the alphabet, x, y, z, w, ...

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knowns

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are denoted by letters at the beginning, a, b, c, d, ...

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Solving the Equation

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The process of expressing the unknowns in terms of the knowns is called

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a solution or root of the equation

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In an equation with a single unknown, a value of that unknown for which the equation is true is called

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solution to the system

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is an assignment of values to all the unknowns so that all of the equations are true. also called set simultaneous equations.

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identity

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a distinction is made between the equality sign ( = ) for an equation and the equivalence symbol () for an

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constants

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Letters from the beginning of the alphabet like a, b, c... often denote

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variables

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letters from the end of the alphabet, like ...x, y, z, are usually reserved for the

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Any real number can be added to both sides. Any real number can be subtracted from both sides. Any real number can be multiplied to both sides. Any non-zero real number can divide both sides. Some functions can be applied to both sides.

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If an equation in algebra is known to be true, the following operations may be used to produce another true equation:

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reflexive relation

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is a binary relation on a set for which every element is related to itself, i.e., a relation ~ on S where x~x holds true for every x in S. For example, ~ could be \"is equal to\".

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An irreflexive, or anti-reflexive

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...

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a binary relation R over a set X is symmetric

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if it holds for all a and b in X that if a is related to b then b is related to a.

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real number

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a value that represents a quantity along a continuum, such as -5 (an integer), 4/3 (a rational number that is not an integer), 8.6 (a rational number given by a finite decimal representation), √2 (the square root of two, an algebraic number that is not rational) and π (3.1415926535..., a transcendental number).

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number line or real line

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Real numbers can be thought of as points on an infinitely long line where the points corresponding to integers are equally spaced called the

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The real number system

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can be defined axiomatically up to an isomorphism

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change of variables

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is a basic technique used to simplify problems in which the original variables are replaced with new ones; the new and old variables being related in some specified way.

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difference of two squares, or the difference of perfect squares

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is a squared (multiplied by itself) number subtracted from another squared number. It refers to the identity

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the method of equating the coefficients

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is a way of solving a functional equation of two polynomials for a number of unknown parameters. It relies on the fact that two polynomials are identical precisely when all corresponding coefficients are equal. The method is used to bring formulas into a desired form.