Algebra Rules Review Sheet Page 10
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10
REVIEW OF ALGEBRA
+
-
+
Notice that we have included the endpoints 2 and 3 because we are looking for values of
x
such that the product is either negative or zero. The solution is illustrated in Figure 3.
x
0
2
3
FIGURE 3
3
2
EXAMPLE 20
Solve
x
3x
4x
.
SOLUTION
First we take all nonzero terms to one side of the inequality sign and factor the
resulting expression:
3
2
x
3x
4x
0
or
x x
1 x
4
0
As in Example 19 we solve the corresponding equation
x x
1 x
4
0
and use the
solutions
x
4
,
x
0
, and
x
1
to divide the real line into four intervals
,
4
,
4, 0
,
0, 1
, and
1,
. On each interval the product keeps a constant sign as shown
in the following chart.
Interval
x
x
1
x
4
x x
1 x
4
x
4
4
x
0
0
x
1
x
1
Then we read from the chart that the solution set is
x
4
x
0 or x
1
4, 0
1,
_4
0
1
The solution is illustrated in Figure 4.
FIGURE 4
ABSOLUTE VALUE
The absolute value of a number , denoted by
a
a
, is the distance from to on the real
a
0
number line. Distances are always positive or , so we have
0
a
0
for every number a
For example,
3
3
3
3
0
0
s2
1
s2
1
3
3
In general, we have
12
a
a
if a
0
Remember that if is negative,
a
■ ■
then
is positive.
a
a
a
if a
0
EXAMPLE 21
Express
3x
2
without using the absolute-value symbol.
SOLUTION
3x
2
if 3x
2
0
3x
2
3x
2
if 3x
2
0
2
3x
2
if x
3
2
2
3x
if x
3
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