Algebra Rules Review Sheet Page 9
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REVIEW OF ALGEBRA
9
3
5
and multiply by , we get
2
6
10
, but if we multiply by
2
, we get
6
10
.
Finally, Rule 5 says that if we take reciprocals, then we reverse the direction of an inequal-
ity (provided the numbers are positive).
EXAMPLE 18
Solve the inequality
1
x
7x
5
.
SOLUTION
The given inequality is satisfied by some values of but not by others. To solve
x
an inequality means to determine the set of numbers for which the inequality is true.
x
This is called the solution set.
First we subtract 1 from each side of the inequality (using Rule 1 with
c
1
):
x
7x
4
Then we subtract
7x
from both sides (Rule 1 with
c
7x
):
6x
4
(
1
)
Now we divide both sides by
6
Rule 4 with
c
:
6
4
2
x
6
3
These steps can all be reversed, so the solution set consists of all numbers greater
(
)
2
2
than
. In other words, the solution of the inequality is the interval
,
.
3
3
2
EXAMPLE 19
Solve the inequality
x
5x
6
0
.
SOLUTION
First we factor the left side:
x
2 x
3
0
We know that the corresponding equation
x
2 x
3
0
has the solutions 2 and 3.
The numbers 2 and 3 divide the real line into three intervals:
, 2
2, 3
3,
On each of these intervals we determine the signs of the factors. For instance,
x
, 2
x
2
x
2
0
?
?
Then we record these signs in the following chart:
Interval
x
2
x
3
x
2 x
3
x
2
A visual method for solving Example 19
■ ■
2
x
3
is to use a graphing device to graph the
x
3
2
parabola
y
x
5x
6
(as in Figure 2)
and observe that the curve lies on or below
the -axis when
x
2
x
3
.
Another method for obtaining the information in the chart is to use test values. For
instance, if we use the test value
x
1
for the interval
, 2
, then substitution in
y
2
x
5x
6
gives
y=≈-5x+6
2
1
5 1
6
2
2
The polynomial
x
5x
6
doesn’t change sign inside any of the three intervals, so we
conclude that it is positive on
, 2
.
Then we read from the chart that
x
2 x
3
is negative when
2
x
3
. Thus,
0
x
1
2
3
4
the solution of the inequality
x
2 x
3
0
is
x 2
x
3
2, 3
FIGURE 2
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