PreCalculus Generic Notes
© by Scott Surgent
Linear Equations and Modeling
We start with the definition of equation. An equation is a complete sentence in which
one expression is set equal to another expression. The left side and the right side of an
equation are called expressions. The equal sign “=” is the verb of the sentence.
An equation is a statement that, depending on the values chosen for the variables, is true
or false. For example, 2 x − 5 = 11 is an equation. If x is chosen to be 8, then this
equation is a true statement. For any other value of x, this equation is a false statement.
A linear equation in one variable is an equation that can be reduced to the form
ax + b = 0 , where x is the variable, and a and b are numerical coefficients, with the
condition that a does not equal zero. For example, 2 x − 8 = 0 is a linear equation. So is
the equation 4( x − 1) + 5 = 7 , since it reduces to 4 x − 6 = 0 after normal algebraic
manipulation. The advantage of the simplified form of a linear equation is that its
solution can be found by performing two quick algebraic steps. In summary, any linear
equation of the form ax + b = 0 has the solution x = − ba .
Some equations become linear after algebraic reduction. For example, ( x + 2) 2 = ( x − 1) 2
looks non-linear, but if we foil and collect terms, we get: x 2 + 4 x + 4 = x 2 − 2 x + 1 which
becomes 6 x = −3 after we cancel the squared terms and collect other terms. The final
result is x = − 12 , which you can check.
Fact: Not every equation has a solution! Consider: 2 x + 1 = x + 3 + x . Simplify and see
what happens.
Solving Equations Graphically
Your graphing utility can be used as an aid in locating solutions to an equation. Please
have your calculator’s user manual handy while working this section.
Consider a graph on a standard Cartesian coordinate system. The point(s) at which the
graph crosses the y-axis is (are) called the y-intercept(s), and the point(s) at which the
graph crosses the x-axis is (are) called the x-intercept(s).
Question: A function can have at most one y-intercept. Why?
A solution to any graph is its x-intercepts. This is very important!
A function can have as few as zero solutions, or many solutions. However, despite the
centuries of development in mathematics, locating solutions algebraically is still only
possible for a few special equations. Other times, the algebraic methods are so difficult
that another method is desired. Since the advent of computers and graphing calculators,
locating solutions to difficult equations is now literally as easy as pressing a few keys.
PreCalculus Generic Notes
© by Scott Surgent
As an example, solve x 3 + x − 3 = 0 . We graph x 3 + x − 3 and note where it crosses the
x-axis. We have to zoom in and use our grapher’s “solve” feature, but you should be able
to show that the solution is x = 1.213 (to 3-deciomal place accuracy). To do this
algebraically would be very difficult, believe it or not. Cubics do not factor easily (we
shall see other ways to approach these problems in a later section).
For example, graph y = x 2 + 3 . This graph does not cross the x-axis. Therefore, this
equation has no solutions. Technically, it has no Real solutions (the x-axis is the real
number line). Later in this lesson we will discuss complex numbers as a method of
locating solutions.
In other cases, you may need to zoom in on the x-intercepts and use your calculator’s
“trace” key to find the x-value of the intercept. Also, the TI-82 and 83 calculators have a
handy key sequence: 2nd-CALC-ROOT (or 2nd-CALC-ZERO). Simply follow the
directions on the screen to locate your solutions.
Lastly, these same “tricks” can be used to determine points of intersection of two
graphs (a useful tool for later this course). Enter both functions, and use the trace key
or the 2nd-CALC-INTERCEPT sequence to find the points of intersection.
Complex Numbers
This section introduces us to a larger set of numbers, the Complex Number Field. This
numbering system will allow us to make sense of polynomials, especially quadratics, that
have no real solutions.
We start by defining the number i = − 1 . It is immediately obvious that i is not a Real
number, since no Real number can equal the square root of –1. Put another way, no Real
number squared is negative. The number i is termed “imaginary”, and from this, we
create the field of complex numbers. A complex number is of the form a + bi, where a is
the real portion, and bi the imaginary portion.
We first note some properties of complex numbers:
1.
2.
3.
4.
5.
i 2 = −1 .
To add two complex numbers, just add the real parts and then the imaginary parts.
Just like collecting terms.
To subtract one complex number from another, distribute the negative as usual
and collect terms.
To multiply two complex numbers, FOIL them in the usual manner. Convert the
i 2 to − 1 and collect terms.
Any real number can be written as a complex number in this manner: 4 becomes 4
+ 0i. The real numbers is a subset of the complex numbers.
PreCalculus Generic Notes
© by Scott Surgent
Don’t dismiss complex numbers as some artificial creation of mathematics. At one time,
negative numbers were regarded as nonsense by most people until a use was found for
them. The complex numbering system, first introduced about 400 years ago, follows the
rules of arithmetic and is self-consistent as well as preserving all of the properties of real
numbers, which we use all the time. Complex numbers are used in physics, fluid
dynamics, and fractal geometry. Complex numbers are sometimes called lateral
numbers, which is a better term in my opinion.
Given a complex number a + bi, we form its conjugate by changing the sign in front of the
bi term, getting a – bi. Complex conjugates have the interesting property of multiplying to
a real number. This allows us to perform division on complex numbers: multiply the
numerator and denominator by the conjugate of the denominator. .
At this time you are simply gaining practice with the complex numbers. We will be
using these in our studies very shortly.
Solving Equations Algebraically
Various algebraic methods are used to solve equations. Factoring, extraction of roots and
completing the square are useful for second-ordered equations (those involving a squared
term). The quadratic formula is also useful.
The Quadratic Formula is a general formula used to solve any quadratic equation. It is
derived using the method of completing the square. While it is a useful formula, many
students stumble while using it, or simply copy it wrong. Practice using the QF, but also
practice other methods. The QF is often best reserved as the option of last resort.
Here it is: Given ax 2 + bx + c = 0, the roots are x =
− b ± b 2 − 4ac
.
2a
Unfortunately, there is no “easy” way to factor higher degreed polynomials other than
getting lucky, or through tedious trial and error. In a later chapter we discuss methods for
locating solutions. Even factoring a cubic polynomial can be daunting!
The usual method to solve equations involving radicals is to isolate the radical and then
square both sides. You must always check your results! Often these steps produce
extraneous solutions, which may violate some mathematical laws.
Equations involving absolute value must be broken into two equations by noting a fact:
Given x = a , then x = a or x = −a . For example, if you were given x = 2 , then we could
let x = 2 or x = −2 since both 2 and − 2 = 2 .
As a further example, if we had
2 x + 3 = 9 , then we can rewrite this as 2 x + 3 = 9 or 2 x + 3 = −9 . Solving each of these
gives us x = 3 or x = −6 . There are two solutions in this example.
PreCalculus Generic Notes
© by Scott Surgent
Note that a statement like x = −2 has no solution since it’s impossible to take the absolute
value of something and get a negative result.
Further note: the word “or” indicates that as long as x is one of the results or the other, you
get a true statement. To be technically correct, you always connect disjoint answers using the
word “or”.
Solving Inequalities Algebraically and Graphically
A common relation used in mathematics is the inequality. Specifically, there are four
different types:
1.
2.
3.
4.
The symbol < which stands for “less than.” For example, 2 < 4 is a true
statement, since 2 is less than 4. On the other hand, 2 < 2 is false, since 2 is not
less than 2.
The symbol ≤ which stands for “less than or equal.” For example, 2 ≤ 4 is a true
statement. So is 2 ≤ 2, since 2 is less than or equal to 2.
The symbol > which stands for “greater than.” For example, 4 > 2 is a true
statement, but 4 > 4 is not.
The symbol ≥ which stands for “greater than or equal.” For example, 4 ≥ 2 is a
true statement, as is 4 ≥ 4.
The relations < and > are called strict inequalities, while the relations ≥ and ≤ allow for
possible equality.
When a variable is introduced into the mix, we can solve for it using the normal
techniques of algebra, with two notable exceptions:
1.
2.
If a < b , then multiplying or dividing through this inequality by a negative
number will cause the inequality to switch orientation, hence − a > −b . For
example, we agree that 2 < 3 is true. It is also clear that − 2 > −3 is also true,
requiring the < to switch to a >. Similarly, the ≤ will switch to a ≥, and vice
versa.
If a < b , then reciprocating will force the < to become a >, getting 1a > b1 . For
example, given 2 < 3, it is also true that 12 > 13 .
Otherwise, treat the inequality as usual and proceed with your algebra.
Since there are many possible solutions to an inequality, it is useful to use a number line
to describe the solution set to an inequality. A parenthesis symbol “(” or “)” is used to
denote < or >, meaning the endpoint is not part of the solution set, while brackets “[“ or
“]” denote ≤ or ≥ , meaning the endpoint is a member of the solution set.