4.3 -Solving Rational Equations and Inequalities
Steps for solving rational equations:
1. Get a common denominator for all terms.
2. Give restrictions.
3. Multiply the numerator of every term by the LCD and reduce each term .
4. Solve for the variable and check your restrictions. If your solution is a restricted value, then it is an
extraneous solution.
Extraneous solutions are solutions that result from the process of solving an equation, but are not valid
solutions to the original equation.
Example 1: Solve each rational equation . Check your solutionis against your restrictions.
( 2._ y +3 =2
' 3
\.,
L\
6
- ~-
.
1
- 2
X
It)
r
b.( ~+3x= 1_
7
14
2
)
LCD:
\'-\
Restrictions:
JU~\-0
~~
Solution: ~
0
Solution:
LCD:
Restrictions: r \ ~\(
~
c.
)
2
- - = -1
X
\ -
f
"'
LCD:
Restrictions:
}
-j 0
92
Unit 4
H Secondary 3
£
10
2x 2x + 5
e. - 2- +
= -x..!..-1- X - 1
X +1
I
LCD:
L\ \~1 1~)
Restrictions:
~ =f
-
:-\)
I
LCD:
-=3h,_
Restrictions:
ltjt"l -ll~
~ (J~ -\qtp
'-\l~P))
\\ t~'~)
'/.
l
±\
\01 ~~ lr' '4 -~ -5 - 2/ 2.,d sx -s
_,7--Jf-7yx ~ S-_ U?-r 3)c-S
1 C\ ~ (o~ tt\ \p
'U -
'J
.,:Z~i- ~
j
-q~.o
~~~
(o ~
(o\f
Solution:~
f
0
Solution:
031
_2_- y - y2 + 4
y+2 ~-/-4
- ( rL;
, }2--)
L~-l\ -\}1~ .\- 2j ~~at~
l\\j ~ l\
--L\
N
~
'±! ~ ~
-"
LCD:
\V\\...,
\( u-"'\
J L,..ol\
J L-)
Restrictions:
'1'
~~2
:~
SOU
I t'IOn~
2} i t 2
93
Example 2: So lve t he fo llowing rationa l inequa lities. Draw a number line and don't forget the restrictions!!!
a.
x+S
x _ >O
3
'f..
r
Restrictions:
X-iJ
T
b.
)
x2 -4
x+4
--~0
Restrictions:
(~ Lt)\y_- 2\
y- 1 2
T
~
I
c.
( - 0() 1
s)( ~ 1 ~
_1_;:::_
2_
x-3 x+3
\
'1-.
/.
)
~
f. ~
Restrictions:
Solution:
I
X1;
...,
T
r
LJ
)
r
i
f
Lj
So lution:
yI
l
l-
>
2
)(
_,)
,/('
.(})V"
"'·$
-j
t-~
)
. (\
~
...1
Solution:
94