MA100, Fall 2010, PART 1: Precalculus
[4] Linear Equations and Inequalities. Absolute Value
1. Linear equations
Definition
A mathematical expression of the form
ax
+
b
where
a, b
∈
R
is called a linear expression
.
For instance, 3
x

7 is a linear expression with
a
= 3 and
b
=

7
.
Solving
ax
+
b
= 0
,
where
a, b
∈
R
Let
a, b
∈
R
,
and consider the equation
ax
+
b
= 0
.
We have the following:
ax
+
b
= 0
=
⇒
ax
=

b
Now:
(A) if
a
6
= 0 then the equation has an unique solution (i.e. one solution only) which is given by
x
=

b
a
;
(B) if
a
= 0 then there are two possibilities: either
b
= 0 or
b
6
= 0
.
i) If
b
= 0 then any
x
∈
R
is a solution.
This is because in this case our equation
ax
+
b
= 0 becomes 0
·
x
= 0
,
that is 0 = 0.
In other words, we obtain have 0 = 0 no matter what
x
is. But 0 = 0 is always true.
In this case, we say that the equation
ax
+
b
= 0 infinitely many solutions.
ii) If
b
6
= 0 then there is no solution.
This is because in this case the equation
ax
+
b
= 0 becomes 0
·
x
=
b
, or 0 =
b.
Since
b
6
= 0, we cannot find any
x
that would solve the equation.
Example: Solve
(23

23)
x
= 7
.
Answer: The equation reads
0
×
x
= 7
=
⇒
0 = 7
=
⇒
there is no solution (or, equivalently, the solution set is void)
Example: Find
m
∈
R
and
n
∈
R
so that the equation for
x,
where
x
∈
R
,
(3
m

2)
x
= (2
n

6)
has infinitely many solutions.
1
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Answer: An equation of the form
ax
=

b
has infinitely many solutions if
a
=
b
= 0
.
In our case,
since
a
= 3
m

2 and
b
= 2
n

6, this means that we have to impose the conditions:
3
m

2 = 0 and 2
n

6 = 0
Solving the above for
m
we have
3
m

2 = 0
=
⇒
3
m
= 2
=
⇒
m
=
2
3
and solving for
n
we obtain
2
n

6 = 0
=
⇒
2
n
= 6
=
⇒
n
=
6
2
=
⇒
n
= 3
.
So for
m
=
2
3
and
n
= 3, the equation (3
m

2)
x
= (2
n

6) becomes 0
·
x
= 0 and thus it has
infinitely many solutions (any
x
∈
R
is a solution).
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 Fall '10
 a
 Calculus, Linear Equations, Equations, Inequalities, ........., Negative and nonnegative numbers, Mathematical Expression

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