Part II: The Answers
146
Answers
1–100
57.
−∞ −
(
] ∪ { }∪ ∞
[ )
,
,
3
0
2
To solve the inequality 2x
4
+ 2x
3
≥ 12x
2
, begin by setting one side equal to zero. Solve
the corresponding equation and then pick a point from each interval (determined by
the solutions) to test in the inequality.
So from 2x
4
+ 2x
3
≥ 12x
2
, you have 2x
4
+ 2x
3
– 12x
2
≥ 0. Factoring the corresponding
equation so you can solve for x gives you
2
2
12
0
2
6 0
2
3
2 0
4
3
2
2
2
2
x
x
x
x x
x
x x
x
+
−
=
+ −
(
) =
+
− =
(
)(
)
Set each factor equal to zero and solve for x, giving you the solutions x = 0,
x = –3, and x = 2. These values are also solutions to the inequality.
Next, pick a test point from each of the intervals, (–∞, –3), (–3, 0), (0, 2), and (2, ∞),
to see whether the answer is positive or negative. Using x = –10 to check the interval
(–∞, –3) gives you
2 10
2 10
12 10
20 000 2 000 1 200
16 800
4
3
2
(
)
(
)
(
)
,
,
,
,
−
+ −
−
−
=
−
−
=
which is greater than zero.
Using x = –1 to check the interval (–3, 0) gives you
2 1
2 1
12 1
2 2 12
12
4
3
2
( )
( )
( )
−
+ −
−
−
= − −
= −
which is less than zero.
Using x = 1 to check the interval (0, 2) gives you
2 1
2 1
12 1
2 2 12
8
4
3
( )
( )
( )
+
−
= + −
= −
which is less than zero.
Finally, using x = 10 to check the interval (2, ∞) gives you
2 10
2 10
12 10
20 000 2 000 1 200
20 800
4
3
2
( )
( )
( )
,
,
,
,
+
−
=
+
−
=
which is greater than zero.
Therefore, the solution set is −∞ −
(
] ∪ { }∪ ∞
[ )
,
,
3
0
2
.
Note that you could’ve divided the original inequality by 2 to simplify the initial
inequality.