Chapter 4: Polynomial and Rational
Functions
Section 4-7: Radical Equations and
Inequalities

• To solve radical equations, the first step is to isolate the radical on one side of the equation. Then raise each side of the equation to the proper power to eliminate the radical expression.
• The process of raising each side of an equation to a power sometimes produces extraneous solutions.
• These are solutions that do NOT satisfy the original equation. Therefore, it is important to check all possible solutions in the original equation to determine if any of them should be eliminated from the solution set.

• Solve x
= x
−
3
+
3 x
−
3
= x
−
3
( x
−
3 )
2 = x
−
3 x
2 −
6 x
+
9
= x
−
3 x
2 −
7 x
+
12
=
0
( x
−
4 )( x
−
3 )
=
0 x
=
4 , x
=
3

•
• Solve
7
=
3
− 3 x
+
11
7
−
3
= −
3 x
+
11
4
= −
3 x
+
11
−
4
=
3 x
+
11
(
−
4 )
3
−
64
=
= x
+
11 x
+
11
−
75
= x

• Solve x
+
1
=
1
+
2 x
−
12
Square each
( x
+
1
2
) (
1
+ side
2 x
−
12
2
) x
+
1
=
1
+
2 2 x
−
12
+
2 x
−
12
− x
+
12
(
− x
+
12
)
2
=
2
=
(
2
2 x
−
12
2 x
−
12
2
) x
2 −
24 x
+
144
=
4 ( 2 x
−
12 ) x
2 −
24 x
+
144
=
8 x
−
48 x
2 −
32 x
+
192
=
0
( x
−
8 )( x
−
24 )
=
0 x
=
8 works in the original equation.
But x
=
24 does not work.
It is an extraneous solution.

6 x
−
5
>
4
6 x
−
5
>
16
• Solve
6 x
−
5
>
4
6 x
>
21
• When working with inequalities x
>
21
6
=
7
2 involving a radical, the excluded values usually involve entire intervals.
6 x
−
5
≥
0
6 x
≥
5 x
≥
5
6
So since
7
2 is bigger tha n
5
6
, x
>
7
2

• Section 4-7
• Pp255-257
• #12-26 even, 28,29,33
• 34, 47,50