Absolute Value Equations
The absolute value of a number is dened by
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,
Functions f : X Y and g : Y X are inverses if
f (g(y) = y
g(f (x) = x
for all x X and y Y . If f has an inverse, it is often denoted f 1 . However, f 1 does not mean !
Example. f (x) = x3 and g(x) = x1/3 are inverses, s
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
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
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
The slope of the line which passes through the points (x1 , y1 ) and (x2 , y2 ) is
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
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
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
There are various wa
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
The idea is
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
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
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
If n is a positive integer, then
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
Absolute Value and Inequalities
| 47| = 47,
|150| = 150
and |0| = 0.
The minus signs dont cancel; theyre blocked by the absolute value.
Geometrically, the absolute value of a number is its distance from the origin
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,
You also know what this mean if x is a rationa
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,
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
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
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
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
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
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
A rational expression is something of the form
Rational expressions can often be simplied by factoring the top and bottom, then cancelling any
common factors. However, cancelling common factors can cha
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 +
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: