Math Analysis – Precalculus, Sullivan 10th Edition
AppendixA6Day1 1 8/12/2019
Appendix A.6 – Solving Equations – Day 1
An equation in one variable is a statement in which two expressions, at least one containing the variable, are
equal. The expressions are called the sides of the equation. Since an equation is a statement, it may or may
not be true, depending on the value of the variable. The admissible values of the variable are those in the
domain of the variable. Values of the variable that result in a true statement are called solutions, or roots, of
the equation. To solve an equation means to find all the solutions of the equation.
We often write solutions of an equation in set notation. This set is called the solution set of the equation.
For example, the solution set of the equation
016x2=−
is
.4,4−
Some equations have no real solution.
An equation that is satisfied for every value of the variable for which both sides are defined is called an identity.
For example, the equation
1x2x3x2 +++=+
is an identity, since the statement is true for any real number x.
One method for solving an equation is to replace the original equation by a succession of equivalent
equations, equations having the same solution set, until an equation with an obvious solution is obtained.
Procedures that Result in Equivalent Equations
1) Interchange the two sides of the equation:
Replace
x4 =
by
4x =
2) Simplify the sides of the equation by combining like terms, eliminating parentheses, and so on:
Replace
)2x(x47)3x( ++=++
by
2x510x+=+
3) Add or subtract the same expression on both sides of the equation:
Replace
37x2 =−
by
737)7x2( +=+−
4) Multiply or divide both sides of the equation by the same nonzero expression
Replace
4x|xDom,
4x
6
4x
x2 =
−
=
−
by
)4x(
4x
6
)4x(
4x
x2 −
−
=−
−
5) If one side of the equation is 0 and the other side can be factored, then use the Zero-Product Property
and set each factor equal to 0:
Replace
0)2x(x =−
by
0x =
or
02x =−
Solving an Equation
Example 1: Solve the equation
39x4 =−
. Example 2: Solve
1x
4
2
1x
x4
−
=+
−
.
39x4 =−
Domain:
1x|x
9399x4 +=+−
−
−=
+
−
−1x
4
)1x(2
1x
x4
)1x(
12x4 =
4)1x(2
1x
x4
)1x( =−+
−
−
4
12
4
x4 =
42x2x4 =−+
3x =
42x6 =−
2422x6 +=+−
6x6 =
6
6
6
x6 =
1x =
But
1x =
is not in the domain of the
variable. So, there is no real solution.
