Concept Byte: Quadratic Inequalities
For Use With Lesson 4-9
To solve some quadratic inequalities, relate the quadratic expression to 0 and factor. To determine the sign of each factor, use what you know about multiplying positive and negative numbers.
Example 1
Solve each inequality algebraically.
-
2
x
2
−
14
x
<
0
2 , bold italic x squared , minus 14 bold italic x less than 0
2
x
(
x
−
7
)
<
0
Factor
.
2
x
>
0
and
(
x
−
7
)
<
0
,
or
2
x
<
0
and
(
x
−
7
)
>
0
The product is negative, so the two factors
must have
different
signs
.
x
>
0
and
x
<
7
,
or
x
<
0
and
x
>
7
Simplify
.
0
<
x
<
7
No value can be both greater than
7
a
n
d
less than
0.
table with 4 rows and 2 columns , row1 column 1 , 2 x . open , x minus 7 , close . less than 0 , column 2 cap factor , . , row2 column 1 , 2 x greater than 0 , and . open , x minus 7 , close . less than 0 comma or 2 x less than 0 , and . open , x minus 7 , close . greater than 0 , column 2 table with 2 rows and 1 column , row1 column 1 , cap theproductisnegativecommasothetwofactors , row2 column 1 , musthave . different . signs , . , end table , row3 column 1 , x greater than 0 , and , x less than 7 comma or x less than 0 , and , x greater than 7 , column 2 cap simplify , . , row4 column 1 , 0 less than x less than 7 , column 2 cap novaluecanbebothgreaterthan . 7 eh n d . lessthan . 0. , end table
-
2
x
2
−
14
x
>
0
2 , bold italic x squared , minus 14 bold italic x greater than 0
2
x
(
x
−
7
)
>
0
Factor
.
2
x
>
0
and
(
x
−
7
)
>
0
,
or
2
x
<
0
and
(
x
−
7
)
<
0
The product is positive
,
so the two factors must
have the
same
signs
.
x
>
0
and
x
>
7
,
or
x
<
0
and
x
<
7
Simplify
.
x
>
7
or
x
<
0
A value that is greater than both
0
and
7
is always greater than
7.
A value that is less than both
0
and
7
is always less than
0.
table with 4 rows and 2 columns , row1 column 1 , 2 x . open , x minus 7 , close . greater than 0 , column 2 cap factor , . , row2 column 1 , 2 x greater than 0 , and . open , x minus 7 , close . greater than 0 comma or 2 x less than 0 , and . open , x minus 7 , close . less than 0 , column 2 table with 2 rows and 1 column , row1 column 1 , cap theproductispositive . comma . sothetwofactorsmust , row2 column 1 , havethe . sehme . signs , . , end table , row3 column 1 , x greater than 0 , and , x greater than 7 comma or x less than 0 , and , x less than 7 , column 2 cap simplify , . , row4 column 1 , x greater than 7 , or , x less than 0 , column 2 table with 2 rows and 1 column , row1 column 1 , cap avaluethatisgreaterthanboth . 0 ehnd 7 . isalwaysgreaterthan . 7. , row2 column 1 , cap avaluethatislessthanboth . 0 ehnd 7 . isalwayslessthan . 0. , end table , end table
You can use a table to solve inequalities by analyzing the values of y around 0.
Activity
Use a table to find the solutions of
x
2
−
6
x
+
5
<
0
.
bold italic x squared , minus 6 bold italic x plus 5 less than 0 .
|
x
|
y
|
| 0 |
5 |
| 1 |
0 |
| 2 |
−
3
negative 3
|
| 3 |
−
4
negative 4
|
| 4 |
−
3
negative 3
|
| 5 |
0 |
| 6 |
5 |
- What happens to the value of y when
0
≤
x
≤
6
?
0 less than or equal to x less than or equal to 6 question mark
- Does this make sense when you think of the shape of the graph of
y
=
x
2
−
6
x
+
5
?
y equals , x squared , minus 6 x plus 5 question mark Explain.
- What x-values in the table make the inequality
x
2
−
6
x
+
5
<
0
x squared , minus 6 x plus 5 less than 0 true?
- What are the solutions of
x
2
−
6
x
+
5
<
0
?
x squared , minus 6 x plus 5 less than 0 question mark
You can determine the solution of a quadratic inequality based on how many times and where the graph of the related function crosses the x-axis. The graph could open upward or downward, and could intersect the x-axis at 0, 1, or 2 points.
