2-4-2012
Absolute Value Equations
The absolute value of a number is dened by
|c| =
+c
c
if c 0
.
if c < 0
Suppose you want to solve an absolute value equation of the form
|(stu)| = (a number).
Replace |(stu)| with (stu), then take cases.
Example. Solve |x
Word Problems Involving Systems of Linear Equations
Many word problems will give rise to systems of equations that is, a pair of equations like this:
2x + 3y = 10
x 6y = 5
You can solve a system of equations in various ways. In many of the examples below,
7-29-2008
Inverse Functions
Functions f : X Y and g : Y X are inverses if
f (g(y) = y
and
g(f (x) = x
1
for all x X and y Y . If f has an inverse, it is often denoted f 1 . However, f 1 does not mean !
f
Example. f (x) = x3 and g(x) = x1/3 are inverses, s
2-27-2008
Direct and Inverse Variation
y varies directly with x (or: x and y are directly proportional) if there is a constant k such that
y = kx.
(You can put the k on either side that is, you can write x = ky instead as long as you pick one
or the other
3-23-2008
Integer Exponents
Ill discuss some rules for working with integer exponents. Actually, these rules work with arbitrary
exponents, but it is easier to explain why theyre true in this case. So in what follows, all the powers are
assumed to be posi
2-27-2008
Functions and Graphs
A function is a rule which assigns a unique output to each input.
Example. In mathematics, functions are often denoted using notation like the following:
f (x) = x2 .
This says that the name of the function is f , and x deno
3-3-2008
Lines
The slope of the line which passes through the points (x1 , y1 ) and (x2 , y2 ) is
m=
y2 y1
.
x 2 x1
The slope measures the rate at which a line goes up or down as you move to the right. For example, a
line with slope 4 goes up 4 units for
8-4-2012
Word Problems Involving Quadratics
These word problems involve situations Ive discussed in other word problems: The area of a rectangle,
motion (time, speed, and distance), and work. However, these problems lead to quadratic equations. You
can so
5-20-2012
Systems of Linear Equations
A system of linear equations in two variables looks like this:
ax + by = p
cx + dy = q
Such a system may have no solutions, one solution (that is, a single pair (x, y), or innitely many
solutions.
There are various wa
1-24-2010
Linear Inequalities
Solving linear inequalities is very similar to solving equations; the only dierence is that = is replaced
with <, >, , or . Here are some linear inequalities:
2x + 1 < 5
3x 2 8x + 14
2 < 3x + 5 26
1
2
x+4> x+6
2
3
The idea is
7-29-2008
The Natural Logarithm
Let a be a positive number, a = 1, and let x > 0. The logarithm of x to the base a is the number
y = loga x such that ay = x. That is,
ay = x.
y = loga x means
Example. What exponential equation is equivalent to log2 16 = 4
9-25-2014
Quadratic Inequalities
In this section, Ill consider quadratic inequalities. Ill solve them using the graph of the quadratic
function. Ill also look at other inequalities, which Ill solve using sign charts.
A quadratic function is a function of
7-29-2008
Polynomials
A polynomial (in one variable) is something of the form
an xn + an1 xn1 + + a1 x + a0 .
The as stand for numbers, and x is the variable (but you can use any letter for the variable). In other
words, a polynomial is a sum of nonnegati
3-23-2008
Fractional Exponents
If n is a positive integer, then
a1/n
is the nth root of a.
If a is positive, it is the positive number b such that
bn = a.
If a is negative, then:
1. If n is odd, a1/n is the negative number b such that
bn = a.
2. If n is e
1-19-2008
Absolute Value and Inequalities
Examples.
| 47| = 47,
while
|150| = 150
and |0| = 0.
41
41
= .
7
7
The minus signs dont cancel; theyre blocked by the absolute value.
Geometrically, the absolute value of a number is its distance from the origin
7-29-2008
The Exponential Function
If a is a positive number and a = 1, the exponential function with base a is
y = ax .
You know what this means when x is an integer; for example,
a3 = a a a,
and
a4 =
1
.
a4
You also know what this mean if x is a rationa
3-30-2008
Complex Numbers
A complex number is a number of the form a + bi, where a and b are real numbers and i =
i2 = 1). For example, here are some complex numbers:
2 + 3i,
77.5i,
13 7,
54,
1 (so
1+i
.
2
Notice that real numbers are special kinds of com
Solving Absolute Value Inequalities
This is a method for solving inequalities like
|ax + b| > c,
|ax + b| < c,
|ax + b| c,
or |ax + b| c.
(In all of these, assume that a = 0.)
There are many approaches to this; you should choose one and stick to it, to av
4-1-2014
Factoring Polynomials
The opposite of multiplying polynomials is factoring. Why would you want to factor a polynomial?
Let p(x) be a polynomial. p(c) = 0 is equivalent to x c dividing p(x).
Recall that when p(c) = 0, you say that c is a root of p
1-24-2010
Formulas
You often have to solve a formula an equation for a variable. The formula may come from
mathematics, but it can also come from another area (such as business, economics, or science). The following
ideas are often useful in solving for a
2-21-2008
Equations Involving Fractions
By an equation with fractions, Ill mean an equation to solve in which the variable appears in the
denominator of one or more fractions. As youve seen with equations involving number fractions, the natural
approach i
4-13-2008
Quadratic Functions
A quadratic function is a function of the form
f (x) = ax2 + bx + c.
Ive already discussed quadratic functions a little; you know that you can use the graph of a quadratic
function is a parabola. The parabola opens upward if
2-25-2008
Word Problems Involving Fractions
Despite the name, not all of these word problems must be solved using fractions. They can be, but
where possible Ive given solutions which avoid fractions. This keeps things simple.
The rst group involve fractio
8-1-2008
Rational Expressions
A rational expression is something of the form
polynomial
.
polynomial
Rational expressions can often be simplied by factoring the top and bottom, then cancelling any
common factors. However, cancelling common factors can cha
1-13-2008
Linear Equations in One Variable
A linear equation in one variable is an equation involving constants and a single variable which
only occurs to the rst power. The following equations are linear equations in one variable:
2x + 1 = 3
4x 5 = 7x +
2-8-2010
The Quadratic Formula
The quadratic formula is a formual for nding the roots of a quadratic equation. It depends on a
procedure called completing the square. Heres the idea. Suppose you can do algebra to get your equation
to look like this:
(vari