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\begin{document}
\chapter{Beginning Algebra Review}
\section{The Real Number Line}
\section{Graphing in the Cartesian Plane}
\section{The Absolute Value}
\newpage
\section{Linear Equations in One Variable}
One of the main goals of algebra is to be able to solve equations.
To {\bf solve} an equation we must find the {\bf solution set},
that is, the set of all values that will make the equation true.
\begin{exa}
Find the solution sets for the follow equations.
\end{exa}
\noindent a) $x+4=9$
{\bf SOLUTION:} If x=5 then x+4=9 becomes the true equation 5+4=9,
hence the solution set is $\{5\}$.
\newline
\newline
b) $x+1=x+2$
{\bf SOLUTION:}There is no number that is the same whether it is
added to one or two, so in this case the solution set is the {\bf
empty set}, written as $\{ \, \, \}$ or $\emptyset$.
\newline
\newline
c) $x^2=4$
{\bf SOLUTION:}The value 2 is clearly in the solution set, since
$2^2=4$, and -2 is also a solution since $(-2)^2=4$, so the
solution set is $\{ -2,2 \}$.
\newline
\newline
d) $x+3=x+3$
{\bf SOLUTION:} Here {\it any real value} will make this equation
true, so the solution set is all real numbers. $\Box$
\newline
These examples demonstrate that the number of solutions might be
anywhere from zero to infinity. As we focus on different types of
equations we will see that we may be able to guess the number of
solutions based on the type of equation. In this section we will
look at linear equations, the most basic type of equation.
\begin{defn}
A {\bf linear equation} in one variable is an equation that can be
written in the form $$Ax+B=0$$ where A and B are real numbers and
$A\neq 0$.
\end{defn}
Linear equations are sometimes called {\bf first order} equations
because the variable is not squared or cubed but only taken to the
first power. It is important to notice what {\it doesn't} happen
in a linear equation.
\begin{exa}
Which of the following equations are linear equations in one
variable?
\end{exa}
a) $3x-12=0$
b) $10y-50=2y+30$
c) $2w+14$
d) $8y=80$
e) $2a+4b=120$
f) $\sqrt{x}+4=0$
g) $\frac{1}{x+1}+\frac{1}{x}=\frac{1}{12}$
h)$\frac{1}{3}x+\frac{2}{5}=13$
\newline
\newline
\noindent {\bf SOLUTION:} Equations a, b, d, and h are the linear
equations. Notice that c is not an equation since there is no
equal sign. Equation e has two variables, a and b. Equation f is
disqualified because it has a square root, and g is not linear
because there is an x in the denominator. Notice, however, that we
{\it can} have fractions in a linear equation, since $\frac{1}{3}$
and $\frac{2}{5}$ are real numbers. $\Box$
\newline
Notice that equation b and equation d have the exact same solution
set, namely $\{10 \}$. Two such equations are called {\bf
equivalent}. Probably it was not immediately clear that 10 was a
solution for equation b, but it is easier to see that it is a
solution to equation d. In general our strategy for solving
equations is to use the rules of algebra to change an equation
like d to an equivalent but simpler equation like b, or better yet
the even simpler (but still equivalent) equation $y=10$. The
following two rules provide valuable tools for simplifying linear
equations.
\newline
\newline
\begin{tabular}{|l|}\hline
\\
{\bf {\large The Addition and Multiplication Properties for
Equations}}
\\
\\For any real numbers A, B, and C
\\
$A=B$ is equivalent to $A+C=B+C$ \\
$A=B$ is equivalent to $A
\cdot C=B \cdot C$ as long as $C \neq 0$
\\ \hline \end{tabular}
\\
\\`
Informally, these properties say that we can add the same number
to both sides of the equation or multiply both sides of the
equation by the same number. Notice that if C happens to be a
negative number then we are adding a negative number to both
sides, which is the same as subtracting. For that reason we could
state a similar "Subtraction Property." Similarly, the
Multiplication Property implies we can also {\it divide} both
sides by any number except zero.
\newline
\begin{tabular}{|l|}\hline
\\
{\bf {\large Strategy for Solving Linear Equations}}\\
\\
1. Combine like terms. (You may have to combine \\ terms again later.)\\
\\
2. Use the Addition Property to gather the terms with the
variable x on \\ one side of the equation and the numbers on the
other side.\\
\\
3. Use the Multiplication Property to eliminate the coefficient
\\ in
front of x.\\
\\
4. Check your answer.\\ \hline \end{tabular}
\begin{exa}
Solve: $2x-19=6-5x+2x$
\end{exa}
\noindent {\bf SOLUTION:} $2x-19=6-3x \qquad Combine\; like\;
terms.$
$2x-19+3x=6-3x+3x \qquad \quad Apply \; the\; Addition\;
Property.$
$5x-19=6 \qquad \qquad \qquad \qquad Combine\; like\; terms.$
$5x-19+19=6+19 \qquad \qquad \quad Apply \; the\; Addition\;
Property,\; again.$
$5x=25 \qquad \qquad \qquad \qquad \qquad Combine\; like\; terms.$
$\frac{5x}{5}=\frac{25}{5} \qquad \qquad \qquad Apply \; the\;
Multiplication\; Property.$
$x=5 \qquad \qquad \qquad \qquad \qquad \qquad Divide.$
{\it Check: } $$ 2 \cdot (5) -19 = ? \; 6-5 \cdot (5)+2 \cdot
(5)$$
$$10 - 19 = ? \; 6-25+10$$
$$ -9 = ? \; -9 \qquad \qquad True! \qquad \Box$$
\newline
In the next example we must first apply the distributive property.
\begin{exa}
Solve: $ $
\end{exa}
When solving a linear equation involving fractions, it is
generally easier if we first multiply each side of the equation by
the least common denominator (LCD) to eliminate the fractions.
When we multiply by the LCD we was making use of the
Multiplication Property.
\begin{exa}
Solve: $ $
\end{exa}
When we solve problems involving interest rates or percentages we
will often work with linear equations with decimal coefficients.
It is often easier in this case to multiply both sides of the
equation by 100.
\begin{exa}
Solve: $ $
\end{exa}
For all of the examples so far the solution set has contained
exactly one value. Such equations are called {\bf conditional
equations} due to the fact that whether or not they are true
depends on the condition that x must equal that one value in the
solution set. While this is the most common case, it turns out
there are two other possibilities. It may happen that the equation
is actually a {\bf contradiction} so that there are no values
which will make it true. In the other extreme the equation may be
an {\bf indentity}, an equation that is {\it always} true
regardless of which real value x is assigned. This last example
demonstrates these possibilities.
\begin{exa}
Solve: $ $
\end{exa}
\end{document}
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